Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
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Evaluation of $\int\frac{x^7+2}{\left(x^2+x+1\right)^2}dx$
Evaluation of $$\int\frac{x^7+2}{\left(x^2+x+1\right)^2}dx$$
$\bf{My\; Try::}$ Let $$\displaystyle \mathop{I = \int\frac{x^7+2}{(x^2+x+1)^2}}dx = \int\frac{(x^7-1)+3}{(x^2+x+1)^2}dx$$
$$\mathop{\displaystyle = \int\frac{x^7-1}{(x^2+x+1)^2}}+\displaystyle \in... | HINT: prove that $$\frac{x^7+2}{(x^2+x+1)^2}={x}^{3}-2\,{x}^{2}+x+2+{\frac {-4\,x-2}{{x}^{2}+x+1}}+{\frac {x+2}{
\left( {x}^{2}+x+1 \right) ^{2}}}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1402786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find the sign of $a,b,c$ in $ax^2+bx+c$ given the graph and a coordinate on it.
So my first approach was that, we see that there are $2$ roots. And one is negative and one is positive. $a$ would be evidently positive. The positive one's modulus is bigger than the negative one's. So the sum of roots would be $+ve$, th... | Assuming that you understand some Calculus!
Let $f(x) = ax^2 + bx + c$ (a second degree polynomial in the single variable $x$) where $a,\ b,\ c$ and $x$ denote real numbers where $a \neq 0$.
Using the Calculus: We can see where some of the given information is coming from and then using it to determine the answer to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1403991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving a rational equation with multiple and nested fractions
This is the equation to solve: $\dfrac{\dfrac{x+\dfrac{1}{2}} {\dfrac{1}{2}+\dfrac{x}{3}}}{\dfrac{1}{4}+\dfrac{x}{5}}=3$
What I did:
$x+\dfrac{1}{2}=\dfrac{2x+1}{2}$
$\dfrac{x}{3}+\dfrac{1}{2}=\dfrac{2x+3}{6}$
$\dfrac{x}{5}+\dfrac{1}{4}=\dfrac{4x+5}{20}$
... | You have, essentially,
$$
\frac{120x+60}{(2x+3)(4x+5)} = 3
$$
Multiply both sides by $(2x+3)(4x+5)$ to get
$$
120x+60 = 3(2x+3)(4x+5) = 24x^2+66x+45
$$
Subtract $120x+60$ from both sides to get
$$
24x^2-54x-15 = 0
$$
Divide both sides by $3$ to get
$$
8x^2-18x-5 = 0
$$
which factors as
$$
(4x+1)(2x-5) = 0
$$
to yield $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Discrete mathematics question $$(n+1)^2+(n+2)^2+(n+3)^2+\dots+(2n)^2=\frac{n(2n+1)(7n+1)}{6}$$
Prove the statement using mathematical induction.
| When $n=1$, $LHS=4=RHS.$
Assume, when $n=k$, $LHS=RHS$. When $n=k+1$,
$LHS=(k+2)^2+(k+3)^2+\dots+(2k+1)^2 + (2(k+1))^2
\\=k(2k+1)(7k+1)/6-(k+1)^2+(2k+1)^2+(2(k+1))^2
\\=[k(2k+1)(7k+1)+18(k+1)^2+6(2k+1)^2]/6
\\=[(2k+1)(7k^2+13k+6)+18(k+1)^2]/6
\\=[(2k+1)(7k+6)(k+1)+18(k+1)^2]/6
\\=(k+1)(14k^2+37k+24)/6
\\=(k+1)(2(k+1)+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1405580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$ If $x^4+7x^2y^2+9y^4=24xy^3$,show that $\frac{dy}{dx}=\frac{y}{x}$
I tried to solve it.But i got stuck after some steps.
$x^4+7x^2y^2+9y^4=24xy^3$
$4x^3+7x^2.2y\frac{dy}{dx}+7y^2.2x+36y^3.\frac{dy}{dx}=24x.3y^2\frac{dy}{dx}+24y^3$
$\dfrac{dy}{dx}=\dfrac... | $$x^4+7x^2y^2+9y^4=24xy^3$$
Can be rewritten as,
$$1+7\frac{y^2}{x^2}+9 \frac{y^4}{x^4}=24\frac{y^3}{x^3}$$
Differentiate wrt $x$, you will get that three things can be equated to $0$. One of them will lead to your result.
(Hint: Don't expand derivative using quotient rule.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1406677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Understanding how to use $\epsilon-\delta$ definition of a limit I finally understand the geometric intuition behind the $\epsilon-\delta$ definition of a limit, which is actually quite neat:
But I'm having trouble actually using the definition to come to a conclusion.
For (a solved) example, to prove that $\lim_{n\ri... | Part (1) Essentially whoever wrote the proof wants to get an $n$ in the denominator (to be explained shortly). Note that $\frac{11}{2(2n+7)}=\frac{11}{4n+14}<\frac{11}{4n}<\frac{3}{n}$, so that we got a nice bound with $n$ in the denominator.
Part (2) Note that we have $|a_{n} - \frac{3}{2}| < \frac{3}{n}$ but really w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1407328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Convert between parameteric ellipse equations I have the parametric equation of an ellipse in this form:
$$x(t)= a\cos(t)$$
$$y(t)=b\cos(t+\phi)$$
It's an ellipse centred about the origin, with a tilt angle. So three parameters.
How can I convert it to the form:
$$x(t)=A\cos(t)\cos(\Phi)-B\sin(t)\sin(\Phi)$$
$$y(t)=A\c... | So we have a pair of equations
$$
a \cos (t+\tau) = A \cos t \cos \Phi - B \sin t \sin \Phi\\
b \cos (t+\tau+\phi) = A \cos t \sin \Phi + B \sin t \cos \Phi
$$
which need to be equal for every $t \in \mathbb R$. Since both of the formula produce an ellipse when $t$ runs over $[u, u+2\pi]$ for any $u$, the value $\tau$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solving $a!+b!+c!=3^d$ The question is to find all tuples $(a,b,c,d)$ of natural numbers $c\geq b$ and $ b \geq a $ and $a!+b!+c!=3^d$. I am finding difficulty in establishing relation between $a$, $b$, $c$, and $d$. I see that a!,b!,c! are even.How can their sum be odd? Please help.
| When $a \le b \le c$ we get
$$
a! + b! + c! = a! \Big( 1 + b!/a! \big( 1 + c!/b! \big) \Big) = 3^d. \tag 1
$$
If $a \ge 2$ then there are no solutions.
If $b \ge 3$ then there are no solutions.
So we can only get
$$
1 + 2 + c! = 3^d, \tag 2
$$
thus
$$
c! = 3 \big( 3^{d-1} - 1 \big) \tag 3
$$
But $3$ divides $c!$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Proving uniqueness of solutions to $\sin^2A + \sin^2B = \sin (A+B)$ without using multivariable calculus In the course of solving a trigonometric problem (see $a^2+b^2=2Rc$,where $R$ is the circumradius of the triangle.Then prove that $ABC$ is a right triangle), in one approach the following equation needed to be solve... | Here are some illustrations, with $\triangle ABC$ having circumdiameter $1$ (and therefore sides $\sin A$, $\sin B$, $\sin C$):
Acute $C$: $\quad \sin C < \sin^2 A + \sin^2 B$
Obtuse $C$: $\quad \sin C > \sin^2 A + \sin^2 B$
Right $C$: $\quad \sin C = \sin^2 A + \sin^2 B$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1409775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Partial fractions problem help I need help with the following:
$\frac{1}{(x-1)^2(x+1)}$
Using the "cover-up" method, I can identify the numerator of the fraction with the denominator $(x+1)$ fairly quickly, and it is $1/4$. Then I arrive at the following: $$1 \equiv (1/4)(x-1)^2 + B(x-1)(x+1)+(Dx+E)(x+1)$$
Then I have ... | Let $\displaystyle \frac{1}{(x-1)^2\cdot (x+1)} = \frac{A}{(x-1)}+\frac{B}{(x-1)^2}+\frac{C}{(x+1)}$
$\displaystyle 1 = A(x-1)\cdot (x+1)+B(x+1)+C(x-1)^2...........(1)$
Put $(x-1) = 0\Rightarrow x=1\;,$ in above equation. So we get $\displaystyle 1 = 2B\Rightarrow B = \frac{1}{2}$
Similarly Put $(x+1) = 0\Rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1410484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find all pair of cubic equations Find all pair of cubic equations $x^3+ax^2+bx+c=0$ and $x^3+bx^2+ax+c=0$, where $a,b$ are positive integers and $c$ not equal to $0$ is an integer, such that both the equations have three integer roots and exactly one of those three roots is common to both the equations.
I tried the sum... | We may assume $a>b\geq1$. An $x$ that solves both equations is $\ne0$ and also solves the equation
$$(a-b)(x^2-x)=0\ .$$
This implies $x=1$ and entails $c=-1-a-b$. Deflating the polynomials in question by the factor $x-1$ leaves us with the pair of equations
$$\left.\eqalign{x^2+(a+1)x+a+b+1&=0\cr y^2+(b+1)y+a+b+1&=0\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1412817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
The differential equation $\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;$ I am learning differential equations and can do the basic examples. However, how can you solve the differential equation
$$\frac{dy}{dx} = \frac{y}{x} - \frac{1}{y}\;?$$
| $$\frac { dy }{ dx } =\frac { y }{ x } -\frac { 1 }{ y } \;$$ substitute
$$y=xt$$
$$ \frac { dy }{ dx } =t+x\frac { dt }{ dx } \\ t+x\frac { dt }{ dx } =t-\frac { 1 }{ xt } \\ x\frac { dt }{ dx } =-\frac { 1 }{ xt } \\ \int { tdt } =\int { -\frac { d }{ { x }^{ 2 } } } \\ \frac { { t }^{ 2 } }{ 2 } =\frac { 1 }{ x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1414760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
In the triangle $ABC$, if $a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$, prove that $3\cdot\widehat{C}=2\cdot\widehat{B}$. Just like in the title, I have to prove that if in a triangle $ABC$
$$a=\frac{2(b^2-c^2)}{-b+\sqrt{b^2+4c^2}}$$
holds, then $3\cdot\widehat{C}=2\cdot\widehat{B}$.
The denominator of the big fraction l... | Assuming $\widehat{ C}=2\theta, \widehat{B}=3\theta,\widehat{A}=\pi-5\theta$ and $R=1$ (the circumradius) we have:
$$ a = 2\sin(5\theta),\quad b=2\sin(3\theta),\quad c=2\sin(2\theta) $$
and $\frac{b^2-c^2}{a}=2\sin\theta$. On the other hand,
$$ \sin^2(\theta) + \sin(3\theta)\sin(\theta)-\sin^2(2\theta)=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1416966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Trying to show $|\overrightarrow{a}\times\overrightarrow{b}|^2=|\overrightarrow{a}|^2|\overrightarrow{b}|^2-(\overrightarrow{a}⋅\overrightarrow{b})^2$
If $\overrightarrow{a} = \langle a_1, a_2, a_3 \rangle$ and $\overrightarrow{b} = \langle b_1, b_2, b_3 \rangle$, then the cross product of $\overrightarrow{a}$ and $\o... | Having grouped your terms together as
$$a_3^2 b_2^2 + a_2^2 b_1^2 + a_2^2 b_3^2 + a_3^2 b_1^2 + a_1^2 b_3^2 + a_1^2 b_2^2 - 2 a_2 a_3 b_3 b_2 - 2 a_1 a_3 b_3 b_1 - 2 a_1 a_2 b_2 b_1$$
we can add and subtract some extra terms (highlighted in colour below):-
$$\begin{align}&a_3^2 b_2^2 + a_2^2 b_1^2 + a_2^2 b_3^2 + a_3^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1418422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Confusing probability problems based on product rule and combinations I am going thru probability exercise. Faced first problem:
Book Q1. Ten tickets are numbered 1,2,3,...,10. Six tickets are selected at random one at a time with replacement. What is the probability the largest number appearing on the selected ticket... | With the actual answers already given, I am putting different possible scenarios and there solutions here for reference.
*
*Selecting six out of {1,2,3,...,10} with replacement. Probability of largest selected number is 7 = ?
Sol.
$$
\overbrace
{\frac{7^6-6^6}{10^6}}
^{
\begin{pmatrix}
\text{no. of ways of selectin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1420232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Solving the infinite series $1-\frac{2^3}{1!}+\frac{3^3}{2!}-\frac{4^3}{3!}+\cdots$ I have the following question:
Evaluate the infinite series:
$$S=1-\frac{2^3}{1!}+\frac{3^3}{2!}-\frac{4^3}{3!}+\cdots$$
(a) $\displaystyle\frac1e$ (b) $\displaystyle\frac{-1}e$ (c) $\displaystyle\frac{2}e$ (d) $\displaystyle\fr... | Looking at the given identity:
\begin{align}
(n+1)^3 &= [n(n-1)(n-2)+6n(n-1)+7n+1] \\
&= n(n^2 - 3n + 2) + 6 n^2 + n + 1 \\
&= n^3 + 3 n^2 + 3 n + 1 = (n+1)^3
\end{align}
Now, consider differentiation of the exponential series:
\begin{align}
D \, e^{t} &= \sum_{n=0}^{\infty} \frac{1}{n!} \, \frac{d}{dt} \, t^{n} =
\su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1423107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Calculating $\int_0^1 \frac{\log (x) \log \left(\frac{1}{2} \left(1+\sqrt{1-x^2}\right)\right)}{x} \, dx$ How would you like to calculate this one? Do you see a fast, neat way here? Ideas?
$$\int_0^1 \frac{\log (x) \log \left(\frac{1}{2} \left(1+\sqrt{1-x^2}\right)\right)}{x} \, dx$$
Sharing solutions is only optional.... | Here is an alternative solution.
\begin{align}
\int^1_0\frac{\ln{x}\ln\left(\frac{1+\sqrt{1-x^2}}{2}\right)}{x}\ {\rm d}x
&=\frac{1}{4}\int^1_0\frac{\ln{x}\ln\left(\frac{1+\sqrt{1-x}}{2}\right)}{x}\ {\rm d}x\tag1\\
&=\frac{1}{4}\int^1_0\frac{\ln(1-x)}{1-x}\ln\left(\frac{1+\sqrt{x}}{2}\right)\ {\rm d}x\tag2\\
&=\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1424130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 1
} |
Find 2 sums with the binomial newton Find the sum of:
i)$\displaystyle\sum_{k=0}^{n} k^2$ $\left(\begin{array}{c} n\\k\end{array}\right)$
ii) $\displaystyle\sum_{k=1}^{n} \frac{2k+5}{k+1}$$\left(\begin{array}{c} n\\k\end{array}\right)$
Thoughts:
i)(After the Edit)$\displaystyle\sum_{k=0}^{n} k^2$ $\left(\begin{array}... | Hint:
ii) We can expand as follows
\begin{align*}
\sum_{k=1}^n\frac{2k+5}{k+1}{n\choose k}&=\sum_{k=1}^n\frac{(2k+2)+3}{k+1}{n\choose k}\\[4pt]
&=\sum_{k=1}^n 2{n\choose k}+\frac{3}{n+1}\sum_{k=1}^n\frac{n+1}{k+1}{n\choose k}\\[4pt]
&=2\sum_{k=1}^n{n\choose k}+\frac{3}{n+1}\sum_{k=1}^n{{n+1}\choose {k+1}}
\end{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Minimal polynomial and algebraic multiplicy
Let $A\in \mathbb{R}^{4 \times 4}$ and the Minimal polynomial $m_A(x)=x^3-2x^2$
What are the eigenvalues and the multiplicy?
$m_A(x)=x^3-2x^2=x^2(x-2)$ so $\lambda_1=0$ and $ \lambda_2=2$ so we now look at the option of the characteristic polynomial we know that it must be ... | Because $x^{2}(x-2)$ is the minimal polynomial, then the Jordan canonical form can be built from the following blocks along the diagonal:
$$
\left[2\right], \left[0\right], \left[\begin{array}{cc} 0 & 1 \\ \cdot & 0\end{array}\right]
$$
You must have the first and third blocks, which, together, occupy ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proof by induction on n So I aim to prove that $n^2 \leq 2^n + 1$ for all integers $n \geq 1$. We can see that this is true for $n=1$ since $1 \leq 3$. Now I suppose that this is true for an arbitrary $k$ such that $k \geq 1$. So $k^2 \leq 2^k +1$. From this, I want to reach a statement $(k+1)^2 \leq 2^{k+1} +1$ by alg... | Suppose that for some $n\ge3$, we have $n^2\le 2^n+1$. Then
$$
\begin{align}
(n+1)^2
&=n^2+2n+1\tag{1}\\
&=2n^2-2-(n+1)(n-3)\tag{2}\\
&\le2n^2-2\tag{3}\\
&\le2^{n+1}\tag{4}\\
&\lt2^{n+1}+1\tag{5}
\end{align}
$$
Explanation:
$(1)$: expand the square
$(2)$: $n^2+2n+1=2n^2-2-(n^2-2n-3)$
$(3)$: since $(n+1)(n-3)\ge0$ for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1428373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Prove that the family of curves $\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$,where $\lambda$ is a parameter,is self orthogonal. Prove that the family of curves $\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$,where $\lambda$ is a parameter,is self orthogonal.
I tried to find the differential equation of first... | Let us find a differential equation not involving $\lambda$ that describes the situation. Then we show that it is invariant under the change of $y'\mapsto -\frac{1}{y'}$.
Differentiating,
$$
\frac{2x}{a^2+\lambda}+\frac{2yy'}{b^2+\lambda}=0.
$$
Then, using the equation and the differentiated one, we find that
$$
a^2+\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1431742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
What are better approximations to $\pi$ as algebraic though irrational number? I know that $\pi \approx \sqrt{10}$, but that only gives one decimal place correct. I also found an algebraic number approximation that gives ten places but it's so cumbersome it's just much easier to just memorize those ten places.
What's a... | Dalzell's integral is related to the rational approximation $\pi\approx \frac{22}{7}$.
$$\pi=\frac{22}{7}-\int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx\approx\frac{22}{7}$$
Similar small integrals are related to simple irrational approximations using $\sqrt{2}$ and $\sqrt{3}$.
$$\pi=\frac{20\sqrt{2}}{9}-\frac{2\sqrt{2}}{3} \int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1432729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 9,
"answer_id": 4
} |
How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$? In case the question didn't display in the title correctly: How do we bound the function $\frac{-1}{2}\sum_{m=n}^{\infty}\frac{1}{m^{2}}$?
I think a way I can do this is to show that $$\sum_{m=n}^{\infty}\frac{1}{m^{2}} < \frac{1}{n^{2}} + \... | Estimating via the integral works well. Another way is to note that the tail (if we forget about the $-\frac{1}{2}$ in front) is less than
$$\frac{1}{(n-1)(n)}+\frac{1}{(n)(n+1)}+\frac{1}{(n+1)(n+2)}+\cdots.$$
But $\frac{1}{(n-1)(n)}=\frac{1}{n-1}-\frac{1}{n}$ and the next term is $\frac{1}{n}-\frac{1}{n+1}$, and the n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1434486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\int \frac{\sqrt{1+x^2}}{1-x^2}dx$ Problem :
$\int \frac{\sqrt{1+x^2}}{1-x^2}dx$
My approach :
Put $x = \tan\theta$
we get $$\int \frac{\sqrt{1+x^2}}{1-x^2}dx = \frac{\frac{\sin^2\theta + \cos^2\theta}{\cos^2\theta \cos\theta}}{\frac{\cos^2\theta -\sin^2\theta}{\cos^2\theta}} d\theta $$
$$= \frac{1}{(\cos^2\theta ... | HINT:
$$\int\dfrac{dy}{(\cos^2y-\sin^2y)\cos y} =\int\dfrac{\cos y\ dy}{(1-2\sin^2y)(1-\sin^2y)}$$
Set $\sin y=u$
Use Partial Fraction Decomposition,
$$\dfrac1{(1-2u^2)(1-u^2)}=\dfrac A{(\sqrt2)^2-u^2}+\dfrac B{1-u^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1435235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Does the series $\sum\frac{(-1)^n cos(3n)}{n^2+n}$ converge absolutely? $\sum|\frac{(-1)^n \cos(3n)}{n^2+n}| \le \sum\frac{1}{n^2+n}$
since $-1 \le cos(3n) \le 1$
$\sum\frac{1}{n^2+n} = \sum\frac{1}{n(n+1)} = \sum(\frac{1}{n} - \frac{1}{n+1})$
$\int(\frac{1}{x} - \frac{1}{x+1}) dx = \log|x| + \log|x+1| + C$
which diver... | Of course, the absolute value of $(-1)^n\cos(3n)$ is $\leq 1$ and the series $\sum_{n\geq 1}\frac{1}{n(n+1)}$ is absolutely convergent (to $1$). Moreover, since:
$$ \sum_{n\geq 1}\frac{x^n}{n(n+1)}=\sum_{n\geq 1}\frac{x^n}{n}-\sum_{n\geq 1}\frac{x^n}{n+1}=1-\left(1-\frac{1}{x}\right)\log(1-x)\tag{1}$$
by taking $x=-e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1439373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Find $n$, where its factorial is a product of factorials I need to solve $3! \cdot 5! \cdot 7! = n!$ for $n$.
I have tried simplifying as follows:
$$\begin{array}{}
3! \cdot 5 \cdot 4 \cdot 3! \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3! &= n! \\
(3!)^3 \cdot 5^2 \cdot 4^2 \cdot 7 \cdot 6 &= n! \\
6^3 \cdot 5^2 \cdot 4^2 \... | If the formula is true it can only possibly be $8!$, $9!$, or $10!$ because $11!$ and larger have a factor of $11$. $8!$ doesn't have a large enough power of $3$, and $9!$ doesn't have a large enough power of $5$, so if the formula holds it must be $10!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1440482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 7,
"answer_id": 2
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Evaluate the limit of ratio of sums of sines (without L'Hopital): $\lim_{x\to0} \frac{\sin x+\sin3x+\sin5x}{\sin2x+\sin4x+\sin6x}$ Limit to evaluate:
$$\lim_{x \rightarrow 0} \cfrac{\sin{(x)}+\sin{(3x)}+\sin{(5x)}}{\sin{(2x)}+\sin{(4x)}+\sin{(6x)}}$$
Proposed solution:
$$
\cfrac{\sin(x)+\sin(3x)+\sin(5x)}{\sin(2x)+\sin... | HINT:
Using Prosthaphaeresis Formula,
$$\sin(a-2)x+\sin(ax)+\sin(a+2)x$$
$$=\sin(ax)+[\sin(a-2)x+\sin(a+2)x]$$
$$=\sin(ax)[1+2\cos2x]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1441163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Integral involving cube root and seventh root Find the value of $$\int_{0}^{1} (1-x^7)^{\frac{1}{3}}-(1-x^3)^{\frac{1}{7}}\:dx$$
My Approach:
Let $$I_1=\int_{0}^{1} (1-x^7)^{\frac{1}{3}}dx$$ and
$$I_2=\int_{0}^{1} (1-x^3)^{\frac{1}{7}}dx$$
For $I_1$ substitute $x^7=1-t^3$ so $dx=\frac{-3t^2}{7}(1-t^3)^{\frac{-6}{7}}\:... | We can write it as $$I = \displaystyle \int_{0}^{1}\sqrt[3]{1-x^7}dx-\int_{0}^{1}\sqrt[7]{1-x^3}dx$$
Now Using $$\displaystyle \bullet \int_{a}^{b}f(x)dx = -\int_{b}^{a}f(x)dx$$
So we get $$I = \displaystyle \int_{0}^{1}\left(1-x^7\right)^{\frac{1}{3}}dx+\int_{1}^{0}\left(1-x^3\right)^{\frac{1}{7}}dx$$
Now Let $$\di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1444591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Integration of $1/((1+\cos x)(1+\sin x))$ How do you integrate $\displaystyle \frac 1{(1+\cos x)(1+\sin x)}$? I've tried $u$ substitution and manipulation and have got nowhere. I cannot think of any other methods that would work.
| Let $$\displaystyle I = \int\frac{1}{(1+\sin x)\cdot (1+\cos x)}dx = \frac{1}{2}\int\frac{1}{\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^2\cdot \cos^2 \frac{x}{2}}dx$$
Above we used $$\displaystyle \bullet\; 1+\sin x = \sin^2 \frac{x}{2}+\cos^2 \frac{x}{2}+2\sin \frac{x}{2}\cdot \cos \frac{x}{2}$$
And $$\displaystyl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How do I evaluate $ \lim_{x \to \pi/2}\frac {\sin x-1}{(1+\cos 2x)}$without using L'Hospital's rule? How do I evaluate this: $$\lim_{x \to \pi/2}\displaystyle \frac {\sin x-1}{(1+\cos 2x)}$$? without using L'Hospital's rule.
Attempt : I used trigonomeitric transformation and I used Taylor series I got this :
$$\lim_{x... | $$\lim_{x\to \frac{\pi}{2}} \frac {\sin x-1}{(1+\cos 2x)} = \lim_{x\to \frac{\pi}{2}} \frac {\sin x-1}{(1+\cos^2x-\sin^2x)} = \\
= \lim_{x\to \frac{\pi}{2}} \frac {\sin x-1}{(1+(1-\sin^2x)-\sin^2x)} = \\
= \lim_{x\to \frac{\pi}{2}} \frac {\sin x-1}{2(1-\sin^2x)} = \\
= \lim_{x\to \frac{\pi}{2}} \frac {\sin x-1}{2(1-\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $b^{2} (\cot A + \cot B) = c^{2}(\cot A + \cot C)$ In any triangle $ABC $ prove that $$b^{2} (\cot A + \cot B) = c^{2}(\cot A + \cot C)$$
How we can prove this trigonometric identity. I tried many ways and use the other well known identity but it wasn't work. My question is how we can prove this trigonometric ... | There is a nice proof of this trig identity.
Notice, in right $\triangle ABC$ $(A+B+C=180^\circ)$ we know from sine rule $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$$
$$\implies \ a=k\sin A, \ b=k\sin B, \ c=k\sin C, $$
Now, we have $$LHS=b^2(\cot A+\cot B)$$
$$=(k\sin B)^2\left(\frac{\cos A}{\sin A}+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1445753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Proving this trig identity:$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$
$$\frac{1+\cos\theta+\sin\theta}{1+\cos\theta-\sin\theta}=\frac{1+\sin\theta}{\cos\theta}$$
What I've tried,
$$\frac{((1+\cos\theta)+(\sin\theta))((1+\cos\theta)+(\sin\theta))}{(1+\cos\theta-\sin\theta... | Multiplying and dividing by $(1+\sin\theta -\cos \theta)$ yields
$$\frac{(1+\sin \theta)^2-\cos^2 \theta}{1-\cos^2\theta -\sin^2\theta +2\sin\theta \cos \theta}= \frac{1+\sin^2 \theta + 2\sin \theta -\cos^2 \theta}{2\sin \theta \cos \theta}$$
Substituting $-\cos^2 \theta = \sin^2 \theta -1$, we obtain
$$ \frac{1+\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1446213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving $\lim_{x \to 4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}$ without L'Hopital. I am trying to solve the limit
$$\lim_{x \to 4}\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}$$
Without using L'Hopital.
Evaluating yields $\frac{0}{0}$. When I am presented with roots, I usually do this:
$$\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}} \cdot \frac{3+\... | $$\frac{3-\sqrt{5+x}}{1-\sqrt{5-x}}=\frac{\frac{x-4}{-(3-\sqrt{5+x})}}{\frac{x-4}{1+\sqrt{5-x}}}=\frac{1+\sqrt{5-x}}{-(3+\sqrt{5+x})}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1447028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How do I prove the existence of $\lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^3+y^3}$ How do I prove the existence of
$$\lim_{(x,y)\to(0,0)} \frac{x^2y^2}{x^3+y^3}$$
When $y=0$ the limits is $0$
when $y=-x$ it´s $$\lim_{x\to0} \frac{x^2(-x^2)}{x^3- x^3}=\lim_{x\to0} \frac{x^2(-x^2)}{0}$$
I can concluir that the limit no exist?... | Along the path $y=x$,
\begin{align*}
\frac{x^2y^2}{x^3+y^3} = \frac{1}{2}x\rightarrow 0.
\end{align*}
However, along the path $y=x^3-x$,
\begin{align*}
\frac{x^2y^2}{x^3+y^3} = \frac{x^8-2x^6+x^4}{x^9+3x^5-3x^7},
\end{align*}
which does not have a finite limit when $x\rightarrow 0$.
That is,
\begin{align*}
\lim_{(x, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1448006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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diagonalization exp(A) counter exemple We consider a finite vector space E over a field K. We consider u in End(E). We can show that if the characteristic polynomial of A splits then we have the following equivalence : A diagonalizable iff exp(A) diagonalizable.
I am searching a counter exemple in the case of a non sp... | trying to guess what you might be asking, take
$$ A =
\left(
\begin{array}{rr}
0 & \pi \\
- \pi & 0
\end{array}
\right).
$$
Then
$$ e^A =
\left(
\begin{array}{rr}
-1 & 0 \\
0 & -1
\end{array}
\right).
$$
If
$$ C =
\left(
\begin{array}{rr}
0 & 1 \\
- 1 & 0
\end{array}
\right)
$$
then $C^2 = -I$ and, for real $t,$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1449970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evalutating $\lim_{x\to0}\frac{1-\cos x}{x^2}$ $$\lim_{x\to0}\frac{1-\cos x}{x^2}$$
I know there are many ways to calculate this. Like L'Hopital. But for learning purposes I am not supposed to do that. Instead, I decided to do it this way:
Consider that $\cos x = 1- \sin^2 \frac{x}{2}$ (from the doulbe-angle formulas h... | Notice, your mistake $\cos x\ne 1-\sin^2\frac{x}{2}$
Now, there are various methods to find the limit here given two methods as follows
Method-1$$\lim_{x\to 0}\frac{1-\cos x}{x^2}$$
$$=\lim_{x\to 0}\frac{2\sin^2\left( \frac{x}{2}\right)}{x^2}=\lim_{x\to 0}\frac{1}{2}\frac{\sin^2\left( \frac{x}{2}\right)}{\left( \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1452958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
$\left[\csc^2\frac{\alpha}{2}+\csc^2\frac{\beta}{2}+\csc^2\frac{\gamma}{2}\right]=2$ If $\cos \alpha \cos \beta \cos \gamma-\cos \alpha-\cos \beta-\cos \gamma+1=0$ and $\alpha\neq\beta\neq\gamma\neq2n\pi$,then prove that $\left[\csc^2\frac{\alpha}{2}+\csc^2\frac{\beta}{2}+\csc^2\frac{\gamma}{2}\right]=2$
$\cos \alpha ... | HINT:
As $\csc^2\dfrac x2=\dfrac2{1-\cos x},$
Please simplify
$$\dfrac2{1-\cos\alpha}+\dfrac2{1-\cos\beta}+\dfrac2{1-\cos\gamma}=2$$ to reach at the given condition
| {
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"url": "https://math.stackexchange.com/questions/1453291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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How to solve $\lim _{x\to \infty \:}\left(\sqrt{x+\sqrt{1+\sqrt{x}}}-\sqrt{x}\right)$ Solved I stuck in this limit, I tried to solve it and gets 1/2 as result. Yet, I was wrong because I forgot a square. Please need help!
$\lim _{x\to \infty \:}\left(\sqrt{x+\sqrt{1+\sqrt{x}}}-\sqrt{x}\right)$
Note: it's $+\infty$
Tha... | If we have $f(x)/x^{1/2} \to 0,$ where $f(x) > 0,$ then the MVT gives
$$(x+f(x))^{1/2} - x^{1/2} = 1/(2c_x^{1/2})\cdot f(x) \le f(x)/2x^{1/2} \to 0.$$
Above $c_x \in (x,x+f(x)),$ so we're OK. In this problem $f(x) = (1+x^{1/2})^{1/2},$ so the desired limit is $0.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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trigonometry equation, $[\cos(2x)]^2-\sin(2x)=1$ I tried to solve this equation: $[\cos(2x)]^2-\sin(2x)=1$
I got different answer from the book.
the answer in the book: $x=-45+180k$ , $x=90k$
Am I right? $x = -45+180k$ is equal to $x= 135+180k$ ? How can I check if it's the same? Or maybe I did a mistake? please help
\... | $$\cos²(2x)-\sin(2x) = 1$$
$$-\sin(2x)= 1-\cos²(2x)$$
$$-\sin(2x)=\sin²(2x)$$
$$0= \sin²(2x)+\sin(2x)$$
$$\sin(2x)(\sin(2x)+1)=0$$
Then, you get two equations:
$$\sin(2x)=0$$
and
$$\sin(2x)+1=0$$
$$\sin(2x)=\sin (180)$$
and
$$\sin(2x)=\sin(270)$$
From the $$\sin(2x)=\sin(180)$$, you can get the solution:
$$2x=180+k.3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1456748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluating the magnitude. Is this less than 1? Where $|x|<1$, I'm looking to determine if
$$\left|\frac{1}{x}(-1+\sqrt{1-x^2})\right|<1$$
I believe it is, since we can use a Taylor series to approximate
$$\sqrt{1-x^2} = 1 - \frac{1}{2}x^2-\frac{1}{4}(x^2)^2 + O(3) \approx 1 - \frac{1}{2}x^2$$
This then gives
$$\left|\f... | Hint: $$
\left|\frac{\sqrt{1-x^2}-1}{x}\right| = \left|-\frac{x}{\sqrt{1-x^2}+1} \right| \leq \frac{|x|}{1}=|x|.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1459304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Prove using factor theorem. Using factor theorem, show that $a+b$,$b+c$ and $c+a$ are factors of
$(a + b + c)^3$ - $(a^3 + b^3 + c^3)$
How do we go about solving this ?
Thanks in advance !
| Let,$f(a)$=$(a+b+c)^3-(a^3+b^3+c^3)$
If $(a+b)$ is a factor,$f(-b)$ must be $0$ by factor theorem.
Putting $f(-b)$,we get,
$(-b+b+c)^3-(-b^3+b^3+c^3)$
$=c^3-c^3=0$.
Similarly prove the others.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find $\lim\limits_{x~\to~4}\frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}}$ I am wondering how to find $\lim\limits_{x~\to~4}\dfrac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt{2}}.$
I have found out that it equals $\dfrac{0}{0}$ if the equation is not simplified.
I have tried multiplying by the conjugate of the numerator (and the... | $$\lim_{x\to 4}\frac{\sqrt{2x+1}-3}{\sqrt{x-2}-\sqrt 2}$$
$$=\lim_{x\to 4}\frac{(\sqrt{2x+1}-3)(\sqrt{2x+1}+3)}{(\sqrt{x-2}-\sqrt 2)(\sqrt{x-2}+\sqrt 2)}\cdot \frac{\sqrt{x-2}+\sqrt 2}{\sqrt{2x+1}+3}$$
$$=\lim_{x\to 4}\frac{2x+1-9}{x-2-2}\cdot \lim_{x\to 4} \frac{\sqrt{x-2}+\sqrt 2}{\sqrt{2x+1}+3}$$
$$=\lim_{x\to 4}\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1461552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to solve this simple problem So my younger brother asked me to solve a simple problem, but I'm not sure how to solve this. Any help would be much appreciated.
$$a^3 - 1/ a^3 = 4. $$
Prove that $$a - 1/a = 1$$
| Using $$\displaystyle \left(a-\frac{1}{a}\right)^3=\left(a^3-\frac{1}{a^3}\right)-3\cdot a\cdot \frac{1}{a}\left(a-\frac{1}{a}\right)$$
So we get $$\displaystyle \left(a-\frac{1}{a}\right)^3=4-3\displaystyle \left(a-\frac{1}{a}\right)\;,$$ Now Put $\displaystyle \left(a-\frac{1}{a}\right)=x$
So we get $$\displaystyle ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1462753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find the equation of the locus of the intersection of the lines below Find the equation of the locus of the intersection of the lines below
$y=mx+\sqrt{m^2+2}$
$y=-\frac{ 1 }{ m }x+\sqrt{\frac{ 1 }{ m^2 +2}}$
By graphing, I have got an ellipse as locus : $x^2+\dfrac{y^2}{2}=1$.
The given lines form tangent and normal ... | Solving the system you get
$x = - \frac{m}{{\sqrt {{m^2} + 2} }},y = \frac{2}{{\sqrt {{m^2} + 2} }}$
Now squaring and adding you get
${x^2} + {y^2} = \frac{{{m^2}}}{{{m^2} + 2}} + \frac{4}{{{m^2} + 2}} = {\left( {\sqrt {\frac{{{m^2} + 4}}{{{m^2} + 2}}} } \right)^2}$
which is a circle with center (0,0) and radius $r = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1466618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Extremal values of $(x+3)^\frac{1}{3} - x^\frac{1}{3}$ $f(x) = (x+3)^\frac{1}{3} - x^\frac{1}{3}$
I am trying to find the extremal values of $f$. I start by differentiating, equating the derivative to 0, and solving that for x:
$f'(x)= \frac{1}{3} ((x+3)^\frac{-2}{3} - x^\frac{-2}{3}) = 0$
$(x+3)^2=x^2$
$x^2 + 6x + 9 =... | I've got it.
$f''(\frac{-3}{2}) < 0$, therefore the extremal is a maximum point.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1467637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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If $t = \tan (x/2)$, find expressions for $\sin x, \cos x$ in terms of $t$. Hence, solve the equation $3\sin x - 4\cos x = 2$.
If $$t = \tan \frac{x}{2},$$ find expressions for $\sin x, \cos x$ in terms of $t$. Hence, solve the equation $$3\sin x - 4\cos x = 2.$$
Attempt:
I have been solving a lot of trig questions l... | If you know the bisection formulas
$$
\left|\sin\frac{x}{2}\right|=\sqrt{\frac{1-\cos x}{2}}
\qquad
\left|\cos\frac{x}{2}\right|=\sqrt{\frac{1+\cos x}{2}}
$$
you can put them together finding
\begin{align}
\left|\tan\frac{x}{2}\right|=\sqrt{\frac{1-\cos x}{1+\cos x}}
&=\sqrt{\frac{1-\cos x}{1+\cos x}\frac{1-\cos x}{1-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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$\int_{-1}^1 e^{-x^{4}}\cdot(1+\ln(x+\sqrt{x^2+1})+5x^3-4x^4)dx$ Is it possible to evaluate this integral $$\int_{-1}^1 e^{-x^{4}}\cdot(1+\ln(x+\sqrt{x^2+1})+5x^3-4x^4)dx$$
I got this question on my math quiz today, Solution not given yet, But is it even possible to evaulate this integral
| Notice that $x\mapsto\ln (x+\sqrt{x^2+1})$ is an odd function since
\begin{align}
\ln [(-x)+\sqrt{(-x)^2+1}]&=\ln\left[\frac{\sqrt{x^2+1}-x}{1}\cdot\frac{\sqrt{x^2+1}+x}{\sqrt{x^2+1}+x}\right]\\
&=\ln\left[\frac{1}{\sqrt{x^2+1}+x}\right]\\
&=-\ln (x+\sqrt{x^2+1})
\end{align}
Also $x\mapsto x^3$ is odd, then we have
\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1468629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Convert advanced parametric equation to regular/cartesian can anybody help me to convert following parametric equation in a form Y =Y(X):
$$
x = cos(t) \sqrt{(2 - cos^2(3t))} \\
y = sin(t) \sqrt{(2 - cos^2(3t))}
$$
I've tried also with Wolfram Alpha and it seem not to work:
Reduce[x == Cos[t] Sqrt[2 - Cos[3 t]^2], {t}... | I would say
$x^2+y^2=(\cos^2t+\sin^2t) (2 - \cos^2(3t)) = 2 - \cos^2(3t)=1+\sin^2(3t)$
$\Rightarrow $equations in polar coordinates:
$r = \sqrt{1+ \sin^2(3t)}=\cdots=\sqrt{1+ \sin^2t(3 \cos^2t-sin^2t)^2}$
and transformation from polar into Cartesian coordinates:
$\displaystyle \,\cos t = \frac{x}{r},\,\, \sin t= \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1470188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Moore-Penrose pseudoinverse of a matrix that is invertible in each column block 4I am not in major in math. But currently I am working with a couple of matrices in the form like this:
\begin{equation}
\left[\begin{array}{rrrrrrrrrrrr}
1 & 0 & 0 & 0 & 0& 0& 0& 0& 0& 0& 0& 0\\
0 & 1 &0.5 &0.5 & 0& ... | Construction of a block pseudoinverse is an unanswered challenge in my research area. It's easy to build block inverses of Toeplitz matrices which are nonsingular.
Here is your data with symmetry axes:
$$
\mathbf{A} =
\left(
\begin{array}{cccccc|cccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If there exist real numbers $a,b,c,d$ for which $f(a),f(b),f(c),f(d)$ form a square on the complex plane.Find the area of the square. For all real numbers $x$,let the mapping $f(x)=\frac{1}{x-i},\text{where} i=\sqrt{-1}$.If there exist real numbers $a,b,c,d$ for which $f(a),f(b),f(c),f(d)$ form a square on the complex ... | Hint: The function $f$ bijectively maps the extended real line into the circle through the origin having centre at $\frac{i}{2}$. The area of a square inscribed in a circle with radius $r$ is $2r^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1473451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
If the biquadratic $x^4+ax^3+bx^2+cx+d=0(a,b,c,d\in R)$ has $4$ non real roots,two with sum $3+4i$ and the other two with product $13+i$ If the biquadratic $x^4+ax^3+bx^2+cx+d=0(a,b,c,d\in R)$ has $4$ non real roots,two with sum $3+4i$ and the other two with product $13+i$.Find the value of $b$.
Since there are four n... | By Vieta’s theorem, $b$ is the sum of all products of pairwise distinct roots, i. e.
$$
\begin{align*}
b &= \alpha\overline\alpha + \alpha\beta + \alpha\overline\beta + \overline{\beta\alpha} + \beta\overline\alpha + \beta\overline\beta\\
&= \overline{(\alpha+\beta)}\cdot (\alpha+\beta) + \alpha\beta + \overline{\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is the odd part of even almost perfect numbers (other than the powers of two) not almost perfect? Let $\sigma(x)$ denote the sum of the divisors of $x$. A number $M$ is called almost perfect if $\sigma(M) = 2M - 1$.
If $M$ is an even almost perfect number, then the only known examples for $M$ are $M = 2^k$, where $k \... |
Is this proof correct?
I have not found any errors in your proof, so it looks correct to me.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1474844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Simplify the fraction with radicals I want to simplify this fraction
$$ \frac{\sqrt{6} + \sqrt{10} + \sqrt{15} + 2}{\sqrt{6} - \sqrt{10} + \sqrt{15} - 2} $$
I've tried to group up the denominator members like $ (\sqrt{6} + \sqrt{15}) - (\sqrt{10} + 2) $ and then amplify with $ (\sqrt{6} + \sqrt{15}) + (\sqrt{10} + 2) $... | HINT :
$$\sqrt 6\pm\sqrt{10}+\sqrt{15}\pm 2=\sqrt 3(\sqrt 5+\sqrt 2)\pm\sqrt{2}(\sqrt 5+\sqrt 2)$$
$$=(\sqrt 5+\sqrt 2)(\sqrt 3\pm\sqrt 2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1476313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Digamma function ratio limit I'd like to know how to go about proving the result
$$\lim_{x\to-n}\dfrac{\psi(x)}{\Gamma(x)} = (-1)^{n-1}n!$$
as it appears on p.g. 2 here http://www.math.usm.edu/lambers/mat415/lecture16.pdf.
| Recall that the analytic continuation of the Gamma function can be obtained by using repeatedly the functional relationship $\Gamma(x+1)=x\Gamma(x)$. Then, for $x+m+1>0$, we can write
$$\Gamma (x)=\frac{\Gamma(x+m+1)}{x(x+1)(x+2)\cdots(x+m-1)(x+m)} \tag 1$$
Using $(1)$ reveals that for $x+m+1>0$, the digamma function ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$2=1$ Paradoxes repository I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one week, I explain the solution in the blackboard and I propose a new one. For example, I poste... | Here's one I just made up. $\log_{b} b^x = x$. And $\log_{b} 1 = 0$.
Let $b = 1, x = 1$, and $b^x = 1$. Then $0 = \log_b 1 = \log_b b^x = x = 1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1480488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 15,
"answer_id": 5
} |
A trigonometry equation: $3 \sin^2 \theta + 5 \sin \theta \cos \theta - 2\cos^2 \theta = 0$
$$3 \sin^2 \theta + 5 \sin \theta \cos \theta - 2\cos^2 \theta = 0$$
What are the steps to solve this equation for $ \theta $?
Because, I am always unable to deal with the product $\sin \theta \cos \theta$.
| Here is a general strategy dealing with these kind of problems. Using summation formulas for sin and cos one can easily prove the following identities
$$\left\{ \matrix{
\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta = 2{\cos ^2}\theta - 1 = 1 - 2{\sin ^2}\theta \hfill \cr
\sin 2\theta = 2\sin \theta \cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1481232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 4
} |
Asymptotic of a sum involving binomial coefficients Could you help me to find an asymptotic for this sum?
$$ \sum_{k=0}^{n - 1} (-1)^k {n \choose k} {3n - k - 1 \choose 2n - k} = {n \choose 0} {3n - 1 \choose 2n} - {n \choose 1} {3n - 2 \choose 2n - 1} + ... + (-1)^{n-1} {n \choose n-1} {2n \choose n + 1} $$
I have ... | Suppose we first seek to evaluate the somewhat more general
$$S_m(n) = \sum_{k=0}^n {n\choose k} (-1)^k
{mn-k-1\choose 2n-k}$$
with $m$ an integer parameter.
Now introduce
$${mn-k-1\choose 2n-k} =
\frac{1}{2\pi i}
\int_{|z|=\epsilon}
\frac{1}{z^{2n-k+1}} (1+z)^{mn-k-1} \; dz.$$
We thus get for the sum
$$\frac{1}{2\pi ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1483351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Calculate limit with a lot of roots $$ \lim_{x\to 0}\frac{\sqrt{1+2x}-\sqrt[\Large3]{1+5x}}{\sqrt[\Large5]{1+x}-\sqrt[\Large5]{1+2x}}$$
Multiplication by conjugate hasn't worked. I need a hint to start.
| Applying L'Hospital's rule, we get
$$\lim_{x\to 0}\frac{\sqrt{1+2x}-\sqrt[3]{1+5x}}{\sqrt[5]{1+x}-\sqrt[5]{1+2x}}$$
$$=\lim_{x\to 0}\frac{\frac{2}{2\sqrt{1+2x}}-\frac{5}{3\sqrt[2/3]{1+5x}}}{\frac{1}{5\sqrt[4/5]{1+x}}-\frac{2}{5\sqrt[4/5]{1+2x}}}$$
$$=\frac{\frac{2}{2}-\frac{5}{3}}{\frac{1}{5}-\frac{2}{5}}=\frac{\frac{-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1483433",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding the second Eigenvector of a repeated Eigenvalue The matrix $A= \left(\begin{matrix} 2 & -3 & 6\\0 & 5 & -6\\0 & 1 & 0\end{matrix}\right)$ for which I am trying to find the Eigenvalues and Eigenvectors.
I have repeated Eigenvalues of $\lambda_1 = \lambda_2 = 2$ and $\lambda_3 = 3$.
After finding the matrix subs... | Let's solve $(A - 2I) v = 0$ for vectors $v$.
The equation resolves to:
$$ \begin{pmatrix} 0 & -3 & 6 \\ 0 & 3 & -6 \\ 0 & 1 & -2 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = 0$$
which imposes the condition $v_2 = 2 v_3$, with no condition on $v_1$. The general solution is, for scalars $a$ and $b$:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1483971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding all positive integer solutions for $x+y=xyz-1$ How do I manually solve $x+y=xyz-1$ assuming that $x, y$ and $z$ are positive integers? I was able to guess all possible solutions, but I do not know how to show that these are the only ones:
$x=1, y=1, z=3$
$x=1, y=2, z=2$
$x=2, y=1, z=2$
$x=2, y=3, z=1$
$x=3, y=2... | Alternative solution:
$x\mid y+1$ and $y\mid x+1$, so $x-1\le y\le x+1$. Simply check cases:
*
*If $x-1=y$, then $y\mid x+1$ gives $x-1\mid x+1$, i.e. $x-1\mid 2$, i.e. $x\in\{3,2\}$, so $y\mid x+1\in\{4,3\}$, so $y\in\{1,2,3,4\}$.
*If $x=y$, then $x\mid 1, y\mid 1$.
*If $x+1=y$, then $x\mid y+1=x+2$, so $x\mid 2$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1484330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
solving the determinants of matrices So i was given this question
Evaluate by first adding all other rows to the first row
My solution:
$
det
\left[ {\begin{array}{cc}
x-1 & 2 & 3 \\
2 & -3& x-2 \\
-2 & x & -2
\end{array} } \right]
$ = $
\left[ {\begin{array}{cc}
x-1 & 2 & 3 \\
2 & -3& x-2 \\
-2 & x & -2
\end{array} }... | No, this is not correct. You may add any row to any other row without changing the determinant. Using your hint you would get
$$
\det\begin{bmatrix}
x-1 & 2 & 3\\
2 &-3 &x-2 \\
-2 & x & -2
\end{bmatrix} = \det\begin{bmatrix}
x-1 & x-1 & x-1\\
2 &-3 &x-2 \\
-2 & x & -2
\end{bmatrix}
$$
Now we can subtract the firs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1488062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Induction proof I'm having trouble with: $1+x+x^2+x^3+...+x^n = \frac{1-x^{n+1}}{1-x}$ So I'm being asked to use induction to prove that for every $x\in\{a\ |\ a\in R, a\neq 1\}$ and for every $n\in N$
$$
1+x+x^2+x^3+...+x^n = \frac{1-x^{n+1}}{1-x}
$$
I have no trouble proving it for $n=1$ :
$$
1+x = \frac{1-x^2}{1-x}
... | From where you are stuck, there is only one little step left: reduce both terms of the LHS to the same denominator. That is, $$\frac{1-x^{k+1}}{1-x} + x^{k+1} = \frac{1-x^{k+1}+(1-x) x^{k+1}}{1-x}=\frac{1-x^{k+1}+x^{k+1} - x\cdot x^{k+1}}{1-x} = \frac{1-x^{k+2}}{1-x}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1488283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit using L'Hopital's Rule I am trying to solve the limit
$$ \lim_{x\to 0}\left(\frac{\frac{1}{x \ln2} - \frac{1}{2^x-1} - \frac{1}2}x\right)$$
using L'Hopital's Rule. However, it seems that I am doing something incorrectly, as after using the rule a couple of times, it does not get me anywhere but only complicates ... | We can proceed as follows
\begin{align}
L &= \lim_{x \to 0}\dfrac{\dfrac{1}{x\log 2} - \dfrac{1}{2^{x} - 1} - \dfrac{1}{2}}{x}\notag\\
&= \frac{1}{2\log 2}\lim_{x \to 0}\frac{2(2^{x} - 1) - 2x\log 2 - x\log 2(2^{x} - 1)}{x^{2}(2^{x} - 1)}\notag\\
&= \frac{1}{2\log 2}\lim_{x \to 0}\dfrac{2(2^{x} - 1) - 2x\log 2 - x\log ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1488459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I solve the following differential equation?
$\frac{dy}{dx}+ \sqrt\frac{1-y^2}{1-x^2}=0$. How do I substitute $y$? Any help would be appreciated thanks.
| $$\frac{dy}{dx}+ \sqrt\frac{1-y^2}{1-x^2}=0$$ or,
$$\frac{dy}{dx}=- \sqrt\frac{1-y^2}{1-x^2}$$ or,
$$\frac{dy}{\sqrt{1-y^2}}=- \frac{dx}{\sqrt{1-x^2}}$$ or,
$$\sin^{-1}y=-\sin^{-1}x+c$$ or,
$$\sin^{-1}y+\sin^{-1}x=c$$
EDIT:
$$\frac{dy}{dx}+ \sqrt\frac{1-y^2}{1-x^2}=0$$ or,
$$\frac{dy}{dx}=- \sqrt\frac{1-y^2}{1-x^2}$$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Find $k$ such that the area is an integer
For some positive integers k, the parabola with equation $y = \frac{x^2}{k} - 5$ intersects
the circle with equation $x^2 + y^2 = 25$ at exactly three distinct points A, B and C.
Determine all such positive integers k for which the area of $\triangle ABC$ is an integer.
P... | Simplifying $x^2+(\frac{x^2}{k}-5)^2=25$ gives
$$x^2(x^2+k^2-10k)=0.$$
In order for this equation to have three distinct real roots, we need to have $10k-k^2\gt 0,$ i.e. $0\lt k\lt 10$.
Then, $$x=0,\pm\sqrt{10k\color{red}{-}k^2}.$$
The area of $\triangle{ABC}$ is given as
$$[\triangle{ABC}]=\frac 12\times 2\sqrt{10k-k^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $ then find $a+b$ $ \lim _{x\rightarrow -\infty }\sqrt {x^{2}+6x+3}+ax+b=1 $
if I use this $$\lim _{x\rightarrow -\infty }\sqrt {ax^{2}+bx+c}=\left| x+\dfrac {b} {2a}\right| $$
I find $a=1,b=4$ but if I try to multiple by its conjugate
$$\lim _{x\rightarrow -\... | Let's start this step by step, $\lim_{x\to -\infty} \sqrt{x^2 + bx + c} + x= \lim_{x\to -\infty} \sqrt{(x + \frac{b}{2})^2 + c - \frac{b^2}{4}} + x = \lim_{x\to -\infty} |x+\frac{b}{2}| +x = -\frac{b}{2}$ Now everything is simpler than ever, $\lim_{x\to -\infty} \sqrt(x^2 + 6x + 3) + x +(a-1)x b = \lim_{x\to -\infty} -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Complex equation The problem says to solve the given complex equation:
$$
z^4-\left[
\frac{\sqrt3}{2}i^{21}+
\frac{\sqrt3}{2}i^{9}+
\frac{8}{(1+i)^6}\right]^9=0
$$
The solution is this:
$$
\cos\left(
\frac{\pi}{8}+\frac{k\pi}{2}
\right)+i\sin\left(
\frac{\pi}{8}+\frac{k\pi}{2}
\right)
,k=0,1,2,3
$$
My problem is that ... | $$z^4-\left(\frac{\sqrt3}{2}i^{21}+\frac{\sqrt3}{2}i^{9}+\frac{8}{(1+i)^6}\right)^9=0\Longleftrightarrow$$
$$z^4-\left(\frac{\sqrt3}{2}i+\frac{\sqrt3}{2}i+\frac{8}{((1+i)^3)^2}\right)^9=0\Longleftrightarrow$$
$$z^4-\left(\frac{\sqrt3}{2}i+\frac{\sqrt3}{2}i+\frac{8}{((1+i)(1+i)^2)^2}\right)^9=0\Longleftrightarrow$$
$$z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1492884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $1+2^n+4^n$ is prime number then prove that $n=3^k$ for some $k\in \mathbb{N}$. If $1+2^n+4^n$ is prime number then prove that $n=3^k$ for some $k\in \mathbb{N}$.
I've looked at https://math.stackexchange.com/a/186723/283318 for an inspiration. In base 2, $1+2^n+4^n=10\ldots010\ldots01$ and I am completely stuck.
| If $n$ is even, then $3\mid 1+2^n+4^n$, so if $1+2^n+4^n$ is prime, then $1+2^n+4^n=3$, so $n=0$.
If $n$ is odd, for contradiction assume $n=3^kr$ for some $r\ge 2,\, \gcd(r,6)=1$.
$$\left(2^n-1\right)\left(4^n+2^n+1\right)=2^{3n}-1=2^{3^{k+1}r}-1$$
$$=\left(2^{3^{k+1}}-1\right)\left(2^{3^{k+1}r-3^{k+1}}+2^{3^{k+1}r-3^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1494706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
How to evaluate the definite integral by the limit definition $\int_{-1}^1 x^3 dx$? Solve the definite integral by the limit definition:
$$\int_{-1}^1 x^3 dx$$
The formula:
$$\int_a^bf(x)dx= \lim_{n\rightarrow \infty} \sum_{i=1}^n f(c_i)\Delta x_i$$
Get the variables:
$$\Delta x_i : \frac{b-a}{n} = \frac{1-(-1)}{n} = \... | When you said
"distribute the cube",
you made a mistake.
(mistake in following expansion corrected)
$(-1+2i/n)^3
=-1+3(2i/n)-3(2i/n)^2+(2i/n)^3
$.
Each term has to be considered.
Also,
note that the function is odd.
What happens if
you take terms
symmetrically located
around the origin?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1497637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to find the remainder $f(x)$=$1+x+x^2+x^3+x^4+x^5$
What is the remainder when $f(x^{12})$ is divided by $f(x)$?
I think remainder theorem cannot be applied (at least directly) here.
I find that $f(-1^{12})$ - $f(-1)$ is $6$ but $f(1^{12})$ - $f(1)$ is $0$
| Hint: It is easier to calculate the remainder of $f(x^{12})(x-1)$ when divided by $f(x)(x-1)=x^6-1$, then divide that remainder by $x-1$ at the end.
Or more directly, note that:
$$x^6\equiv 1\pmod {f(x)}$$
so:
$$x^{12}\equiv 1\pmod{f(x)}$$
So what is $f(x^{12})\pmod{f(x)}$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1499987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Relation between hypergeometric and gamma functions Show that, for a positive integer $n$, $$F\left(-\frac{n}{2},-\frac{n}{2}+\frac{1}{2};n+\frac{3}{2};-\frac{1}{3}\right)=\left(\frac{8}{9}\right)^n\frac{\Gamma\left(\frac{4}{3}\right)\Gamma\left(n+\frac{3}{2}\right)}{\Gamma\left(\frac{3}{2}\right)\Gamma\left(n+\frac{4}... | Your formula still holds if $n\notin \mathbb{Z}$. From $F(a,b;c;z)=(1-z)^{c-a-b}F(c-a,c-b;c;z)$, it suffices to calculate
$$F(\frac{3n+3}{2},\frac{3n+2}{2};n+\frac{3}{2};-\frac{1}{3})$$
its value follows directly from a cubic transformation of $_2F_1$ due to Goursat:
$$\tag{1}_2F_1(\frac{3a}{2}, \frac{3a-1}{2}; a+\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1504934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
If $\phi$ : $F_2[X]/(X^3+X+1)$ $\rightarrow$ $F_2[X]/(X^3+X^2+1)$ is an isomorphism, then prove that $\phi(a)=b+1$, with $a$, $b$ the classes of $X$ Consider two fields:
$K: $ $F_2[X]/(X^3+X+1)$. Let $a$ be the class of $X$ (so $a=X+(X^3+X+1))$
$L: $ $F_2[X]/(X^3+X^2+1)$. Let $b$ be the class of $X$ (so $b=X+(X^3+X^2+1... | Define $[g(X)]$ to be the class of $g(X)$.
There are 8 elements in $F_2[X]/(X^3+X+1):$
$[0],[1],[X],[X^2],[X^3],[X^4],[X^5],[X^6]$.
Note that:$[X^3] = [X+1],
[X^4] = [X^2+X],
[X^5] = [X^2+X+1],
[X^6] = [X^2+1],[X^7] = [1]$
$[X],[X^2],[X^4]$ are roots of $y^3+y+1$
and $[X^3],[X^5],[X^6]$ are roots of $y^3+y^2+1$.
So ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1507610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How can one show that limit of $\frac{1}{1-x}$ as $x$ goes to $2$ exists? How can one show that limit of $\frac{1}{1-x}$ as $x$ goes to $2$ exists?
Its limit value is $-1$.
How can I prove this using epsilon and delta?
| Initially we may guess that
$$
\frac{1}{1-x} \to -1
$$
as $x \to 2$. To prove this, note that we have $x\neq 1$ only if
$$
\bigg| \frac{1}{1-x} - (-1) \bigg| = \bigg|\frac{2-x}{1-x} \bigg| = \bigg|\frac{x-2}{x-1} \bigg|;
$$
we have $0 < |x-2| < 1/2$ only if $||x-1|-1| \leq |x-2| < 1/2$, only if $1/2 < |x-1|$, and only... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1508742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 0
} |
Solving an equation by telling the value of $x^2+y^2$. I have a problem solving an equation. The equation is:
$xy+x+y=44$ and $x^2y+xy^2=448$
and we have to tell the value of $x^2+y^2$
First I tried solving this by doing the following:
$xy+x+y=44~\to~x+y=44-xy~\to~x^2+2xy+y^2=44^2-88xy+x^2y^2~\Rightarrow$
$\Rightarrow~... | Let $A = xy$ and $B = x + y$. Then we have
$$A + B = 44$$
$$AB = 448$$
By Viete's formula, $A$ and $B$ are the roots of
$$r^2 - 44r + 448 = 0$$
$$(r - 16)(r - 28) = 0$$
such that
$$(xy, x + y) = (16, 28),~(28, 16)$$
from which it is trivial do deduce $x^2 + y^2$:
$$\begin{align}x^2 + y^2 &= (x + y)^2 - 2xy
\\&= 28^2 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1518594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
How to solve this limit without using L'Hospital's Rule? $$
\lim\limits_{x\to{\infty}}\frac{\arctan \frac{3}{x}}{|\arctan \frac{2}{x}|}
$$
Can anybody help me to solve this one ?
I ve done somethig like this but im not sure if it is the correct aproach.
$$
\lim\limits_{x\to{\infty}}\frac{\arctan\frac{3}{x}}{|\arctan ... | Since $\arctan{}'(x) = 1/(1 + x^2)$, $arctan'(0) = 1$. Therefore you have
$$\lim_{x\to 0} {\arctan(x)\over x} = 1.$$
This will enable you to do the rest.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1520704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to evaluate the following integral ? What substitution will be helpful? $$\int\frac{cotx }{(1-sinx)(secx+1)}dx $$
we can write this as
$$\int \frac {cosecx.cotx}{(cosecx-1)(secx+1)}dx $$
Now $cosecx=t$ gives $cosecx.cotx=-dt $ which appears in the numerator, what to do about $secx+1$ ?
| $\displaystyle\int\frac{cotx }{(1-sinx)(secx+1)}dx =\int\frac{cos^2x }{sinx(1-sinx)(1+cosx)}dx =\int\frac{1+sinx}{sinx(cosx+1)}dx$
$\displaystyle\int\frac{1+sinx}{sinx(cosx+1)}dx=\int\frac{1}{sinx(cosx+1)}dx+\int\frac{1}{(cosx+1)}dx$
$\displaystyle\int\frac{1}{4sin(x/2)cos^3(x/2)}dx+\int\frac{sec^2(x/2)}{2}dx$
$\displa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1522028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to tell is a matrix is a covariance matrix? How can we know that these matrices are valid covariance matrices?
$$
C= \begin{pmatrix}
1 & -1 & 2 \\
-1 & 2 & -1 \\
2 & -1 & 1 \\
\end{pmatrix}
\\
C= \begin{pmatrix}
4 & -1 & 1 \\
-1 & 4 & -1 \\
... | A square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see here). Positive semi-definite means that
$$
x^{T}Cx\ge0
$$
for every real vector $x$, where $x^T$ is the transpose of the vector $x$.
We have that
\begin{align*}
x^TC_1x
&=\left(\begin{array}{ccc}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1522397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Conics - Locus of points The ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ has one of its foci at the point $F$. The perpendicular from the origin to the tangent at a point $P(a\cos\theta, b\sin\theta)$ on the ellipse intersects the line $FP$ at a point $G$. Find the locus of $G$ as $\theta$ varies.
| HINT - modified:
I would say, the locus will circle:
$FP:\,y=\frac{b\sin \theta}{a\cos \theta - e}\,(x+e)$
$SR:\,y=\frac{a}{b}\,\tan \theta\, x=\frac{a}{b}\,\tan \theta\,(x+e)-\frac{a\,e}{b}\,\tan \theta$
$\Rightarrow x+e = a\,\frac{a \cos \theta-e}{e\cos \theta -a}, \quad y = a\,\frac{b \sin \theta}{e\cos \theta -a}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1524285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $5$ divides $3^{3n+1}+2^{n+1}$
Prove that $5$ divides $3^{3n+1}+2^{n+1}$
I tried to prove the result by induction but I couldn't.
The result is true for $n=1$.
Suppose that the result is true for $n$ i.e $3^{3n+1}+2^{n+1}=5k$ for some $k\in \mathbb{N}$. We study the term
$$3^{3n+4}+2^{n+2}=3^{3n+1}3^3+2^{... | try this
$$3^{3n+1}3^3+2^{n+1}2-3^{3n+1}+2^{n+1}=3^{3n+1}\cdot26+2^{n+1}$$
$$3^{3n+1}\cdot26+2^{n+1}=3^{3n+1}\cdot25+3^{3n+1}+2^{n+1}$$
since $3^{3n+1}+2^{n+1}=5k$
$$3^{3n+1}\cdot25+3^{3n+1}+2^{n+1}=3^{3n+1}\cdot25+5k$$
the difference is divisable by $5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1526409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 6
} |
Maximum value of the integral: $\int _{10}^{19} \frac{\sin x}{1+x^a}dx$
Find the minimum odd value of $a$, where$a>1,~ a \in \mathbb{N}$ such that $$\int_{10}^{19} \frac{\sin x}{1+x^a}dx<\frac{1}{9}$$
ATTEMPT:- Let $I(a)=\int _{10}^{19} \frac{\sin x}{1+x^a}dx$ then from Leibnitz's rule, $$I'(a)=-\int _{10}^{19} \f... | Use the sequence of inequalities (taking into account $a>1$) $$\int _{10}^{19} \frac{\sin x}{1+x^a}dx <\int _{10}^{19} \frac{|\sin x|}{1+x^a}dx<\int _{10}^{19} \frac{1}{1+x^a}dx<\int _{10}^{19} \frac{1}{x^a}dx=\frac{19^{1-a}-10^{1-a}}{1-a}$$ Now, consider $$f(a)= \frac{19^{1-a}-10^{1-a}}{1-a}$$ $$f(2)=\frac{9}{190}<\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1526611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate $ \int_{1}^{\infty} \frac{\sqrt{x - 1}}{(x + 1)^{2}} ~ \mathrm{d}{x} $. I need to solve the following integral:
$$
I = \int_{1}^{\infty} \frac{\sqrt{x - 1}}{(x + 1)^{2}} ~ \mathrm{d}{x}.
$$
Wolfram Alpha gives the answer as $ \dfrac{\pi}{2 \sqrt{2}} $.
I think it’s achievable by complex analysis, but I really ... | Let $u=\sqrt{x-1}$, $du=\frac{1}{2\sqrt{x-1}}$, then
$$I=2\int_0^\infty\frac{u^2}{(u^2+2)^2}du$$
We can apply partial fractions here.
$$=2\int_0^\infty\left(\frac{1}{u^2+2}-\frac{2}{(u^2+2)^2}\right)du$$
$$=2\int_0^\infty\frac{1}{u^2+2}du-4\int_0^\infty\frac{1}{(u^2+2)^2}du$$
The first integrand is almost $\tan^{-1}$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1526717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$? If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$.
Can you give me any hint that I can start with?
| $f(2x)=f^2(x)-2f(x)-\frac{1}{2}
$
and
$f(1) = 2
$.
I can see how to get
$f(2^n)$
and expressions for
$f(2^{-n})$,
but I don't see how to get
$f(3)$.
I show what I've got so far.
All fairly trivial.
$f(2x)+\frac32
=f^2(x)-2f(x)+1
=(f(x)-1)^2
$
so
$f(x)
=1\pm\sqrt{f(2x)+\frac32}
$.
$x = 0
\implies f(0) = f^2(0)-2f(0)-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1528236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Find polynomial $f(x)$ based on divisibility properties of $f(x)+1$ and $f(x) - 1$ $f(x)$ is a fifth degree polynomial. It is given that $f(x)+1$ is divisible by $(x-1)^3$ and $f(x)-1$ is divisible by $(x+1)^3$. Find $f(x)$.
| Note that $(x-1)^3$ divides $f(x)+1$ and $f(-x)-1$, so $(x-1)^3$ divides the sum $f(x)+f(-x)$.
Similarly, we can note that $(x+1)^3$ divides $f(x)+f(-x)$.
Therefore, $(x+1)^3(x-1)^3$ divides $f(x)+f(-x)$, which has degree at most $5$.
This implies that $f(x)$ is an odd function.
We get $f(x)+1=(x-1)^3(ax^2+bx-1)$, (not... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1528699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 2
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Let $a_1, a_2, a_3...$ be the sequence of all positive integers relatively prime to 75. Find the value of $a_{2008}$. Let $a_1, a_2, a_3...$ be the sequence of all positive integers relatively prime to 75, where $a_1<a_2<a_3...$ with $a_1=1, a_2=2, a_3=4, a_4=7$. Find the value of $a_{2008}$.
What I have done:
If ${a_n... | The number of positive integers relatively prime to 75 and less than 75 is 40, as given by Euler's formula.
Thus, in the numbers up to 3750, there are 2000 such numbers. Now we find that the eighth positive integer relatively prime to 75 is 14; hence the answer is 3750+14=3764.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1529844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Number of lines which are normal as well as tangents to the curve $y^2=x^3$? Number of lines which are normal as well as tangents to the curve $y^2=x^3$?
The line passes through two points $(x_1,f(x_1))$ and $(x_2,f(x_2))$ on the curve.
What is the general method to solve such problems?I could'nt proceed much.
| Firstly draw a graph of your curve. You'll see there are two distinct sections to it: $\sqrt{x^3}$ and $-\sqrt{x^3}$. For a line to be a tangent and a normal it must be a tangent to one and normal to the other. Also note due to symmetry if we find one then the vertical reflection must also be an answer so lets look at ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1530128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Basic quadratic calculus Given $$ AX^2+2X-1=0 \,\ \text{where} \,\ A>0$$
What value of A would make the absolute value of both roots bigger than $1$?
I found the roots using quadratic formula and also found that A has to be both $[0,3]$ and $[-1,0]$ and $A>0$, does it mean such an $A$ doesn't exist?
Thanks!
| For $A>0$ we have
\begin{align}
0&=X^2 + 2 \frac{1}{A}X - \frac{1}{A} \\
&= (X + 1/A)^2 -1/A^2 -1/A \\
&= (X - (-1/A))^2 + (- (A+1)/A^2) \iff \\
X &= \frac{\pm\sqrt{A+1}-1}{A}
\end{align}
This is a parabola with vertex at $(-1/A, -(A+1)/A^2)$.
Then
$$
\frac{\sqrt{A+1}-1}{A}
\le \lvert X\rvert
= \left\lvert \frac{\pm\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1531387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Two expressions using three-digit floating point arithmetic with rounding? What is the result of evaluating the following two expressions using three-digit floating point arithmetic with rounding?
$(113. + -111.) + 7.51$
$113. + (-111. + 7.51)$
*
*$9.51$ and $10.0$ respectively
*$10.0$ and $9.51$ respectively
*$9.... | Recall that in $n$-digit floating-point arithmetic numbers are represented by a signed $n$-digit integer and an exponent. Rounding occurs when the significant digits of a number exceed $n$, in which case only the first $n$ are kept and the $n$th is adjusted according to the value of the $(n+1)$th.
You cannot get much m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1531692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Evaluate improper integral $\int_0^\infty \frac{x\sin x}{x^2+1}dx$ How to prove that $$\int_0^\infty \frac{x\sin x}{x^2+1}dx=\frac{\pi}{2e}$$
I've tried several basic approaches like substitution and IBP but can't move forward.
| Suppose we are interested in
$$J = \int_0^\infty \frac{x\sin(x)}{1+x^2} dx$$
In view of the ML bound that we will be using later it proves
convenient to write this as
$$\left[ -\frac{x\cos(x)}{1+x^2} \right]_0^\infty
+ \int_0^\infty \frac{1-x^2}{(1+x^2)^2} \cos(x) dx
= \frac{1}{2}
\int_{-\infty}^\infty \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1531799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Laplace inverse of $\frac{e^{-\sqrt{s+2}}}{s}$ I want to find out $$\mathcal{L^{-1}}\{\frac{e^{-\sqrt{s+2}}}{s}\}$$
How do you find the inverse Laplace?
thanks
| Thank you for the interesting question. Here is a rather brute force solution, which may add a few steps to Jan Erland's solution.
First, let us recall that, if
$$
\mathcal L(f) = \int_0^\infty f(t) \, e^{-st} \, dt = F(s)
$$
Then
$$
\mathcal L(f') = \int_0^\infty f'(t) \, e^{-st} \, dt = s \, F(s) - f(0).
$$
In our ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the limit $\lim_{x \to 0} (2^x + \sin (3x)) ^{\cot(3x)}$ Please help, I have already tried every thing I can, but nothing works. I have no I idea what to do.
$$\lim_{x \to 0} \; (2^x + \sin (3x)) ^{\cot(3x)}$$
| Rewritting the limit using the exponential function $e^x$ using the logarthmic identity $e^{\ln x} = x$.
\begin{align}
\lim_{x \to 0} (2^x + \sin(3x))^{\cot(3x)} &= \lim_{x \to 0} \exp(\cot(3x)\ln(2^x + \sin(3x)))\\
&= \exp( \lim_{x \to 0} \cot(3x)\ln(2^x + \sin(3x)))\\
&= \exp(\lim_{x \to 0} \frac{\ln(2^x + \sin(3x))}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1532657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Most elegant method to calculate $\int^\pi_0 (x\sin x)^2 dx$ $\displaystyle \int^\pi_0 (x\sin x)^2 dx$.
I can easily use integration by parts to solve this integral; however, it is quite messy and I'm just wondering if there exists another alternative method that is more elementary and elegant.
I have tried the substi... | Using the identity $\sin^2(x)=\frac{1}{2}(1-\cos(2x))$, we have:
\begin{align*}
\int^\pi_0 x^2\sin^2 x\; dx &= \frac{1}{2}\int^\pi_0 x^2(1-\cos(2x)) dx \\
&= \frac{1}{2} \int^\pi_0x^2\;dx - \frac{1}{2}\int^\pi_0x^2\cos(2x)\;dx \\
&= \frac{\pi^3}{6} -\frac{1}{2} \left[\frac{1}{2}x^2\sin(2x)\right]_0^\pi + \frac{1}{2}\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1534103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Methods to prove a series is decreasing? What are soome good methods to show a series is decreasing?
Usually I just compare the $a_n$ and $a_{n+1}$ values. For example $\frac 1 {k^2} >\frac 1 {k^2+1}$ so it shows the series is decreasing. I heard you can also do something like $\frac{a_{n+1}}{a_n}$ but that seems compl... | If you are more comfortable with things like $\frac{1}{k^2} > \frac{1}{k^2 +1}$ then another approach for cases like $\frac{k+2}{k(k+3)}$ is to use partial fraction decomposition:
$$\frac{k+2}{k(k+3)} - \frac{k+3}{(k+1)(k + 4)} = \left(\frac{1/3}{k + 3} + \frac{2/3}{k}\right) - \left(\frac{1/3}{k + 4} + \frac{2/3}{k+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1536974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
The trigonometric solution to the solvable DeMoivre quintic? Using the relations for the Rogers-Ramanujan cfrac described in this post,
$$\frac{1}{r}-r = x$$
$$\frac{1}{r^5}-r^5 = y$$
and eliminating $r$ yields,
$$x^5+5x^3+5x = y$$
This is the case $a=1$ of the solvable DeMoivre quintic,
$$x^5+5ax^3+5a^2x+b = 0\tag1$$
... | Thanks to Tito for the nice question. Here is a solution in terms of hyperbolic sine, which may not be what you want.
$$
\sinh(5t) = 5 \sinh t+ 20 \sinh^3 t + 16 \sinh^5 t.
$$
With $x = 2\sinh t$ and $b = -2\sinh 5t$, we have
$$
x^5+ 5 x^3 + 5 x + b = 0\tag1,
$$
which is the $a = 1$ case. So the solution is
$$
x = 2\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1537069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Prove $ 5(a^4+b^4+c^4)+9≥8(a^3+b^3+c^3) $ if $a^2+b^2+c^2=3$ Let $a,b,c\in\mathbb{R}$ such that $a^2+b^2+c^2=3$. Prove that:
$$
5(a^4+b^4+c^4)+9≥8(a^3+b^3+c^3)
$$
I tried to homogenize the inequality to get:
$$
5(a^4+b^4+c^4)+(a^2+b^2+c^2)^2≥\frac{8}{\sqrt3}(a^3+b^3+c^3)\sqrt{(a^2+b^2+c^2)}
$$
I hoped that one don't ne... | By AM-GM:
$$\begin{aligned}a^4+a^4+a^4+a^4+a^4+a^2+a^2+1&\ge 8\sqrt[8]{a^{4\times 5}a^{2\times 2}}\\&=8\sqrt[8]{a^{24}}\\&=8|a^3|\ge8a^3.\end{aligned}$$
Similarly for $b$ and $c$ and add them up.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1537396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How to evaluate $\sin^{-1} (\sqrt{2} \sin \theta) + \sin^{-1} (\sqrt{\cos2 \theta})$ $$\sin^{-1} (\sqrt{2} \sin \theta) + \sin^{-1} (\sqrt{\cos2 \theta})$$
to evaluate the above equation, I used the formula: $$\sin^{-1} (x) + \sin^{-1} (y) = \sin^{-1}[x(1-y^2) + y(1-x^2)]$$
therefore I have got,
$$\sin^{-1} (\sqrt{2} ... | Set $a = \sqrt 2 \sin\theta$ and $b = \sqrt{\cos 2\theta}$. It follows that $a^2 + b^2 = 2\sin^2 \theta + \cos 2\theta = 1$.
If $\alpha = \sin^{-1} a$ where $\alpha \in [-\frac \pi 2, \frac \pi 2]$, we get $\sin(\alpha) = a$, and since $b\ge 0$ we have $b = \sqrt{1 - a^2} = \cos(\alpha) = \sin(\frac \pi 2 - \alpha)$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Division with decimal in the divisor I understand HOW to do division with a decimal in the divisor, but my question is MUST we remove the decimal in the divisor and if so, why?
Thanks.
| Remember that the word decimal is a shortened form of decimal fraction, meaning a fraction whose denominator is a power of $10$. For example,
$$
3.14 = \frac{314}{100},
\qquad
0.0625 = \frac{625}{1000},
\qquad
5280.1 = \frac{52801}{10}.
$$
As an example, consider the division problem
$$
\frac{9.6}{3.84} = \frac{\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1539401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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A new approach to find value of $x^2+\frac{1}{x^2}$ When I was teaching in a college class, I write this question on board.
If we now $\bf{x+\dfrac{1}{x}=4}$ show the value of $\bf{x^2+\dfrac{1}{x^2}=14}$
Some student asks me for a multi idea to show or prove that.
I tried these : $$ $$1 :$$\left(x+\frac{1}{x}\right)... | Not sure how to show what the continued fraction equals but here's my attempt. Again similar in spirit to the algebraic method.
Solving for $x$ in the first equation we have: $x = 4 - \frac{1}{x}$.
Substituting this into the second equation:
$
(4 - \frac{1}{x})^{2} + \frac{1}{(4 - \frac{1}{x})^{2}} = 16 - \frac{8}{4-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 1
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Prove $4(a^2 + b^2 + c^2) + 9(abc)^2 \geq 21.$ Could anyone advise me how to prove this inequality? Hints will suffice, thank you.
Let $a,b,c \in [0, \infty)$ such that $ab+bc+ ca =3.$ Prove $4(a^2 + b^2 + c^2) + 9(abc)^2 \geq 21.$
My attempt: By AM-GM inequality, $(abc)^{2}\leq 1$ and $a^2 + b^2 + c^2 \geq 3(abc)^{\fr... | If we assume the following condition (without loss of generality) $a\geq b \geq c$ then we have :
$$4(a^2+b^2+c^2)+9(abc)^2\geq 4(a^2+b^2+c^2)+9(c)^6$$
With your initial condition $ab+bc+ca=3$ the RHS of the inequality can be rewritten as follows :
$$4((\frac{3-ab}{a+b})^2+(a+b)^2-2ab)+9(\frac{3-ab}{a+b})^6$$
Now we p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find $f$ if $ f(x)+f\left(\frac{1}{1-x}\right)=x $ Find $f$ if
$$
f(x)+f\left(\frac{1}{1-x}\right)=x
$$
I think, that I have to find x that $f(x) = f\left(\frac{1}{1-x}\right)$
I've tried to put x which make $x = \frac{1}{1 - x}$, but this equation has no roots in real numbers.
| If $g(x)=\frac{1}{1-x}$ then verify that $g(g(g(x)))=x$.
Then:
$$\begin{align}
f(x)+f(g(x))&=x\\
f(g(x)) + f(g(g(x))) &= g(x)=\frac{1}{1-x}\\
f(g(g(x))) + f(x) &= g(g(x))=\frac{x-1}{x}
\end{align}$$
So this gives three linear equations, so solve for $f(x)$.
Subtract the second row from the first:
$$f(x)-f(g(g(x)))=x-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Computing the hyperbolic cosine as a serie So I want to prove that:
$$\sum_{x=0}^\infty \frac{\lambda^{2x}}{(2x)!}=\cosh(\lambda)$$
But, proceeding backwards, the only thing I know is that
$$\cosh(\lambda)=\frac{e^{\lambda}+e^{-\lambda}}{2}=\sum_{x=0}^\infty \frac{\lambda^x}{2(x!)}+\sum_{x=0}^\infty \frac{(-\lambda)^x... | $$
\begin{array}{cccccccccc}
& \left(1\vphantom{\dfrac{\lambda^3} 6}\right. & + & \lambda & + & \dfrac{\lambda^2} 1 & + & \dfrac{\lambda^3} 6 & + & \dfrac{\lambda^4}{24} & + & \dfrac{\lambda^5}{120} & + & \cdots \\[10pt]
+ & 1 & - & \lambda & + & \dfrac{\lambda^2} 1 & - & \dfrac{\lambda^3} 6 & + & \dfrac{\lambda^4}{24}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.