Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find the limit: limit x tends to zero [arcsin(x)-arctan(x)]/(x^3) I'm having difficulty in finding the following limit.
$$\lim_{x\to 0}\frac{\arcsin(x)-\arctan(x)}{x^3}$$
I tried manipulating the given limit in standard limit(s) but I got nowhere. I tried L'Hôpital's rule and then realized it would be very lengthy and ... | Without using L'Hospital's and Series Expansion,
Using this and Proof of $\arctan(x) = \arcsin(x/\sqrt{1+x^2})$,
$$\arcsin x-\arctan x=\arcsin x-\arcsin\frac x{\sqrt{1+x^2}}=\arcsin\left(x\cdot\frac1{\sqrt{1+x^2}}-\sqrt{1-x^2}\cdot\frac x{\sqrt{1+x^2}} \right)$$
$$=\arcsin\left(x\cdot\frac{(1-\sqrt{1-x^2})}{\sqrt{1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/760569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Finding the Asymptotic Curves of a Given Surface I have to find the asymptotic curves of the surface given by $$z = a \left( \frac{x}{y} + \frac{y}{x} \right),$$ for constant $a \neq 0$.
I guess that what was meant by that statement is that surface $S$ can be locally parametrized by $$X(u,v) = \left( u, v, a \left( \fr... | I don't have much time for elaborating now, but you definitely can do this yourself (feel free to add your own answer).
Just observe that substitutions $x = r \cos \theta$ and $y = r \sin \theta$ to $z = a \left( x/y + y/x \right)$ turn this equation to $z = \tfrac{a}{\cos \theta \cdot \sin \theta}$ which clearly does... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/761350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Proof that $n$ is prime when $1+x+x^2+...+x^{n-1}$ = ${1-x^n}\over{1-x}$ is prime. Given $x$ and $n$ are positive integers and $1+x+x^2+...+x^{n-1}$ is a prime number.
Then prove that $n$ is a prime number.
Can the formula $1+x+x^2+...+x^{n-1} = \dfrac{1-x^n}{1-x}$ be somehow used?
| An equivalent statement is that if $n = km$ then $\frac {1-x^n} {1-x}$ is also composite. See how you can use the formula $1+x+x^2+...+x^{n-1} = \frac {1-x^n} {1-x}$ to factorize $\frac {1-x^{km}} {1-x}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/761601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove $\cos 3\theta = 4 \cos^3\theta − 3 \cos \theta$ $\cos 3θ = 4 \cos^3 θ − 3 \cos θ$
Here's my attempt. Is it correct? Thanks!
$\cos(3θ)$
$= \cos(2θ + θ)$
$= \cos(2θ)\cos(θ) - \sin(2θ)\sinθ$
$= (2\cos^2θ - 1)\cosθ - (2\sinθ\cosθ)\sinθ$
$= 2\cos^3θ - \cosθ - 2\sin^2θ\cosθ$
$= 2\cos^3θ - \cosθ - 2(1 - \cos^2θ)\cosθ$
$... | Almost good. You made a mistake on the second last line: the $2$ should have been distributed to both the $1$ and the $\cos^2\theta$
Instead of $2\cos^3\theta-\cos\theta-(2\cos\theta-\cos ^3\theta)$, it should be $2\cos^3\theta-\cos\theta-(2\cos\theta-2\cos^3\theta)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/762365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find roots of $3^x+x^3=17$ Find $x$ in the following equation:
$$3^x+x^3=17$$
| Note that if $y\gt 0$we have $3^y\gt 1$ so that if $x\ge 0$ $$3^{x+y}+(x+y)^3=3^x\cdot 3^y +x^3+(3x^2y+3xy^2+y^3)\gt 3^x+x^3$$
And it is very easy to show that for $x\le 0$ we have $3^x+x^3\le 1+0\lt 17$
So the function is increasing for positive $x$ (without calculus) and there is at most a single solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/763354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Non-homogenous System where did I go wrong? Solve the system $\vec{x^{'}}=\begin{pmatrix}2 & -5\\1 & -2 \end{pmatrix}\vec{x}+ \begin{pmatrix} -\cos t\\ \sin t \end{pmatrix}$
The Eigenvalues are $(2-\lambda)(-2-\lambda)+5=0 \implies \lambda=\pm i$
The Eigevalue we choose is $\lambda=i$ and get a fundamental matrix of $$... | Note: the author used $\lambda = i$ and $v_1 = (5, 2-i)$ for the calculations.
You have a sign issue:
$$\phi^{-1}(t)=\dfrac{1}{5}\begin{pmatrix} \cos t-2\sin t & 5\sin t\\ 2 \cos t + \sin t & -5\cos t \end{pmatrix}$$
Look at the sign of entry $\phi^{-1}(t)_{22}$, which should be a negative.
Multiplying by $g(t)$, we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Calculation of $\int_{0}^{1}\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}dx$ Calculation of $\displaystyle \int_{0}^{1}\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}}dx$
$\bf{My\; Try::}$ Let $x=\tan \psi\;,$ Then $\displaystyle dx = \sec^2 \psi$
So Integral convert into $\displaystyle \int_{0}^{\frac{\pi}{4}}\frac{\tan \psi-1+\... | Letting $x=\tan \theta$ gives
$$
\begin{aligned}
\frac{x-1+\sqrt{x^2+1}}{x+1+\sqrt{x^2+1}} &=\frac{\tan \theta-1+\sec \theta}{\tan \theta+1+\sec \theta} \\
&=\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta+1} \\
&=\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}+2 \sin ^2 \frac{\theta}{2}}{2 \sin \frac{\the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/768387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
How Do I Integrate? $\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}$ How do I integrate this one? $$\int \frac{-2x^{2}+6x+8}{x^{2}(x+2)}\,dx$$
Is my answer correct: $$-3\ln\left \| x+2 \right \|+\ln\left \| x \right \|+\frac{4}{x}+C$$
| Express in partial fractions:
$$\frac{8 + 6x - 2x^{2}}{x^{2}(x + 2)} = \frac{4}{x^{2}} + \frac{1}{x} - \frac{3}{x + 2}$$
Integrate term by term:
$$\int \frac{4}{x^{2}} + \frac{1}{x} - \frac{3}{x + 2} \ \mathrm{d}x = -\frac{4}{x} + \ln{|x|} - 3\ln{|x + 2|} + C$$
So you are missing a negative sign.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/768834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the value of $\sin 47^{\circ}+\sin 61^{\circ}- \sin25^{\circ} -\sin11^{\circ}$? After simplification using sum to product transformation equations I keep ending up with
$$4\cos36^\circ\cdot\cos7^\circ\cdot\cos18^\circ$$
How do I simplify this to a single term?
| First we rearrange
\begin{align}
X = \sin 47 + \sin 61 - \sin 25 - \sin 11 & = \sin 47 - \sin 25 + \sin 61- \sin 11
\end{align}
In general, we have $\sin a - \sin b = 2\cos(\frac{a+b}{2})\sin(\frac{a-b}{2})$, and therefore, $\sin 47 - \sin 25 = 2\cos 36 \sin 11$, and $\sin 61 - \sin 11 = 2\cos 36\sin 25$. Substituting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/769103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
If $x\equiv2\pmod{3}$ prove that $3|4x^2+2x+1$ I've tried many different things to get a factor of $k-2$ but keep failing.
If $x\equiv2\pmod{3}$ prove that $3 \mid 4x^2+2x+1$
| Another way to prove it:
Since $x \equiv 2 \mod 3$, $\exists k \in \mathbb{Z}$ such that $x=2+3k$. Therefore:
$$4x^2+2x+1=4(2+3k)^2+2(2+3k)+1=21+54 k+36 k^2 = 3(12k^2+18k+7)$$
So clearly $3 \mid (4x^2+2x+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/769293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Is this a good way to prove that $3^x+4^x =5^x $ has $x=2$ as the only real solution? Divide both sides of the equation $3^x+4^x=5^x$ by $5^x$.
$$ \Rightarrow \frac { 3^x }{ 5^x } +\frac { 4^x }{ 5^x } =\frac { 5^x }{ 5^x }$$
$$\tag1 \Rightarrow \left( \frac 3 5 \right)^x + \left( \frac { 4 }{ 5 } \right)^x =1$$
$\be... | Let $f(x)=(\frac{3}{5})^x+(\frac{4}{5})^x$
This function is strictly decreasing , and as such it's graph will only intersect the line $y=1$ in only one point(suppose there are two points, due to the fact that it is strictly decreasing, these points must coincide).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/774438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solving limit without using L'Hopital $$\lim_{x\to 0} \frac{\ln(\cos x)}{\ln(\cos 3x)}$$
The answer is $1/9$, but I don't want to use L'Hopital because I'm not supposed to.
Any help would be greatly appreciated and thanks in advance.
| Since $\ln(x)=\displaystyle{\sum_{n=1}^{\infty}{\frac{(-1)^{n+1}(x-1)^n}{n!}}}$ and $\cos x=\displaystyle{\sum_{n=0}^{\infty}{\frac{(-1)^nx^{2n}}{(2n)!}}}=1-\frac{x^2}{2}+\frac{x^4}{24}+\ldots$ we can estimate $\ln(\cos x)$ and $\ln(\cos 3x)$ as follows
\begin{align}
\ln(\cos x)&=-\frac{1}{2}x^2+O(x^4) &\text{and} & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/774878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Solve $\cos x+8\sin x-7=0$
Solve $\cos x+8\sin x-7=0$
My attempt:
\begin{align}
&8\sin x=7-\cos x\\
&\implies 8\cdot \left(2\sin \frac{x}{2}\cos \frac{x}{2}\right)=7-\cos x\\
&\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=7-1+2\sin ^2\frac{x}{2}\\
&\implies 16\sin \frac{x}{2}\cos \frac{x}{2}=6+2\sin^2 \frac{x}{2}\\
&\i... | Use the formula for linear combinations of sine and cosine:
$$A \cos x+B \sin x=C \sin (x+\phi) $$ where $C=\sqrt{A^2+B^2},\phi=\arg (A+Bi)$.
This makes transforms your equation to:
$$\sqrt{65} \sin(x+\tan^{-1} 8)=7 $$
which can be easily solved.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$ Hi I am trying to show$$
I:=\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}.
$$
Thank you.
What a desirable thing to want to prove! It is a wo... | As Lucian stated in the comments, integrating by parts shows that the integral is equivalent to showing that $$ \int_{0}^{\infty} \frac{x}{1+x^{2}} \frac{1}{\sinh \frac{\pi x}{2}} \ dx = \frac{\pi}{2}-1 .$$
Let $ \displaystyle f(z) = \frac{z}{1+z^{2}} \frac{1}{\sinh \frac{\pi z}{2}} $
and integrate around a rectangle w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/776679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 4,
"answer_id": 2
} |
Algebra Iranian Olympiad Problem If:
$x^2+y^2+z^2=2(xy+xz+zy)$
and $x,y,z \in R^+$
Prove:
$\frac{x+y+z}{3} \ge \sqrt[3]{2xyz}$
I tried my best to solve this thing but no use.
Hope you guys can help me.Thanks in advance.
| You have:
\begin{align}
(x + y + z)^2
&= x^2 + y^2 + z^2 + 2 (x y + x z + y z) \\
&= 4 (x y + x z + y z) \\
&= 4 \cdot 3 \cdot \frac{x y + x z + y z}{3} \\
&\ge 12 \sqrt[3]{x^2 y^2 z^2} \\
x + y + z
&\ge 2 \sqrt[3]{3 x y z} \\
\frac{x + y + z}{3}
&\ge \sqrt[3]{\frac{8}{9} x y z}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/776931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Simplifying $\frac1{1+x}+\frac2{1+x^2}+\frac4{1+x^4}+\frac8{1+x^8}+\frac{16}{x^{16}-1}$ We need to simplify $$\dfrac{1}{1+x}+\dfrac{2}{1+x^2}+\dfrac{4}{1+x^4}+\dfrac{8}{1+x^8}+\dfrac{16}{x^{16}-1}$$
The last denominator can be factored and we can get all the other denominators as factors of $x^{16}-1$. I tried handling... | $$\frac{16}{x^{16}-1}=\frac{8}{x^8-1}+\frac{-8}{x^8+1}$$
So the $4^{th}$ term of the original sum and the $2^{nd}$ part of the decomposition above are canceled. You are left with:
$$
\dfrac{1}{1+x}+\dfrac{2}{1+x^2}+\dfrac{4}{1+x^4}+\dfrac{8}{x^{8}-1}
$$
Continue similarly. In the end you will get $\frac{1}{x-1}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/777966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
The 'sine and cosine theorem' - formulas for the sum and difference I've read somewhere that the sine and cosine functions can be fully described by this theorem:
*
*$\sin(0) = 0, \cos(0) = 1$
*$\sin(a-b) = \sin(a)\cos(b) - \sin(b)\cos(a)$
*$\cos(a-b) = \cos(a)\cos(b) + \sin(a)\sin(b)$
*There is na $r>0$ such tha... | I created this shortcut proof for the cos addition formula, it uses the cosine and sine formula
The proof I know usually involves complex triangle shapes, so I'll be working with a basic triangle here
$a+b+c = \pi$, and $c = \pi -(a+b)$
$$ C^2 = A^2+B^2-2AB\cos{c}$$
$$ C^2 = A^2+B^2-2AB\cos{(\pi-(a+b))}$$
$$\frac{C^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 2
} |
To find relatively prime ordered pairs of positive integers $(a,b)$ such that $ \dfrac ab +\dfrac {14b}{9a}$ is an integer How many ordered pairs $(a,b)$ of positive integers are there such that g.c.d.$(a,b)=1$ , and
$ \dfrac ab +\dfrac {14b}{9a}$ is an integer ?
| $$\frac ab+\frac{14b}{9a}=\frac{9a^2+14b^2}{9ab}$$
If this expression is an integer, since $a$ and $b$ are coprime, $a$ divides $14$ and $b$ divides $9$.
If $b=1$, we get
$$a+\frac{14}{9a}$$
which is not an integer.
If $b=9$,
$$\frac{a}9+\frac{14}a=\frac{a^2+126}{9a}$$
which is neither an integer, since $a$ can't be a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
Sequence $x_{n+1}=\sqrt{x_n+a(a+1)}$ $a\gt0$, and $x_0=0$, $x_{n+1}=\sqrt{x_n+a(a+1)}, n=0,1,2,\dotsc$.
compute
$$\lim_{n\to\infty}\big(a+1\big)^{2n}\big(a+1-x_n\big)$$
The problem is difficult, I have no idea.
Thank you!
| From the recursion given, we get
$$
x_{n+1}^2=x_n+a(a+1)\tag{1}
$$
Subtracting $(1)$ from $(a+1)^2$ gives
$$
(a+1)^2-x_{n+1}^2=(a+1)-x_n\tag{2}
$$
which is equivalent to
$$
\frac{(a+1)-x_{n+1}}{(a+1)-x_n}
=\frac1{(a+1)+x_{n+1}}\tag{3}
$$
which inductively implies
$$
(a+1)-x_n\le\frac{(a+1)-x_0}{(a+1)^n}\tag{4}
$$
Multi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/779823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Evaluating $\int (\tan^3x+\tan^4x) dx $ using substitution solving $$\int (\tan^3x+\tan^4x) dx $$ using substitution $$t = \tan x$$
My approach has led me to $ \int (1+t)t\sin^2xdt$ which has an $x$ too much and isn't easily solvable for me. If I remove the $x$ I get $\sin^2(\arctan(t))$ and that's not too nice to work... |
unfortunately $\sec$ is not used in my country
If you feel unconfortable with $\sec $ you may proceed from your last integral
\begin{equation*}
I=\int (1+t)t\sin ^{2}\left( \arctan t\right) \,dt
\end{equation*}
by using the following trigonometric identity
\begin{equation*}
\sin ^{2}x=\frac{\tan ^{2}x}{1+\tan ^{2}x},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/780111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
6 keys and a door( probabilities) We have a key ring with 6 keys, but only one key opens the door. We test the keys to the door with no preference until we find the right key. Now we have two different cases:
1) At the first case when a key is not the right one we discard it and try the next one until we find the... | For 1), let X be the number of tries. Then E(X) is the expected number of tries. Recall that $E(X)=\sum_{x=1}^{x=6}xP(X=x)$.
$E(X)=1(\frac{1}{6})+2(\frac{5}{6})(\frac{1}{5})+3(\frac{5}{6})(\frac{4}{5})(\frac{1}{4})+4(\frac{5}{6})(\frac{4}{5})(\frac{3}{4})(\frac{1}{3})+5(\frac{5}{6})(\frac{4}{5})(\frac{3}{4})(\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/781073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How find the maximum value of $|bc|$ Question:
Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$
It is said this is answer is $$|bc|\le \dfrac{3\sqrt{3}}{16}$$
My idea: let $z=1$,then
$$|a+b+c|\le 1$$
let $z=-1$,then... | You can assume that $a,b,c$ are reals and $b,c\ge0$. For $c$ this is obvious. For $b$ you can get this substituting $\theta z$ for $z$, as you wrote in the question. Finally, you can replace the polynomial by $\frac{(az^2+bz+c)+(\bar{a}z^2+bz+c)}2$.
By the maximum principle it suffices to verify the condition on the ci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/781820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Fourier Series Fourier Transform Method I understand $f$ is even about $\pi$ but i'm struggling conceptually with the part I have underlined.
| For completeness, I will work through the entire problem so it can benefit others who view the post. To find, the Fourier coefficients we use
\begin{align}
a_0 &= \frac{1}{2\pi}\int_0^{2\pi}\frac{(\pi - \theta)^2}{4}d\theta\\
&= \frac{\pi^2}{12}\\
a_n &= \frac{1}{\pi}\int_0^{2\pi}\frac{(\pi - \theta)^2}{4}\cos(n\theta)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/783985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How prove $\frac{a}{11a+9b+c}+\frac{b}{11b+9c+a}+\frac{c}{11c+9a+b}\le\frac{1}{7}$ Qustion:
$a,b,c\ge 0$,show that
$$\dfrac{a}{11a+9b+c}+\dfrac{b}{11b+9c+a}+\dfrac{c}{11c+9a+b}\le\dfrac{1}{7}\tag{1}$$
I found this method can't work,
$$x=11a+9b+c,y=11b+9c+a,z=11c+9a+b$$
$$\Longrightarrow a=\dfrac{8x-7y+5z}{126},b=\d... | Let $a=\min\{a,b,c\}$,$b=a+u$ and $c=a+v$.
Hence, we need to prove that
$$49(u^2-uv+v^2)a+(9u+v)(u-3v)^2\geq0$$
Done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/785835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Points on $(x^2 + y^2)^2 = 2x^2 - 2y^2$ with slope of $1$ Let the curve in the plane defined by the equation: $(x^2 + y^2)^2 = 2x^2 - 2y^2$
How can i graph the curve in the plane and determine the points of the curve where $\frac{dy}{dx} = 1$.
My work:
First i found the roots of this equation with a change of variable ... | I would try to solve this as an implicit derivative:
$$\frac{d}{dx}(x^2 + y^2)^2 = \frac{d}{dx}(2x^2 - 2y^2)$$
$$2(x^2+y^2)(2x+2y*y') =4x-4y*y'$$
$$2 x^3+2x^2 y* y'+2 y^3 y'+2 x y^2 = 4x-4y*y'$$
$$2x^2y*y' + 2y^3y' + 4y*y' = 4x-2xy^2-2x^2$$
$$y'(2x^2y + 2y^3+4y) =4x-2xy^2-2x^2$$
$$y' = \frac{4x-2xy^2-2x^2}{2x^2y + 2y^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/785900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
When a quadratic involving three primes is a perfect square How do we find all primes $p,q,r$ such that $p^2+q^2+rpq$ is a perfect square ?
with $r=7$ and $p=q$ we have the expression a perfect square
| Generally before searching primes need to understand what formula solutions of this equation.
Generally speaking, this equation has a lot of formulas for the solution. Because it is symmetrical.
Write the formula can someone come in handy. the equation:
$Y^2+aXY+X^2=Z^2$
Has a solution:
$X=as^2-2ps$
$Y=p^2-s^2$
$Z=p^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/786156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $4xy+3=c^2+3d^2$, is $xy$ necessarily a square? I have a polynomial which, simplified, ends up in the form
$$4xy+3 = c^2+3d^2.$$
Evidently $4xy+3$ is of the form $a^2+3b^2$, in light of the equality. But does
$$
c^2 + 3d^2 = 4xy + 3 = xy(2)^2 + 3(1)^2
$$
necessarily force $xy$ to be a square? I just can't see ho... | For the equation: $4xy+3=c^2+3d^2$
Can be written like such a simple solution:
$x=2a^2+6t^2-6at-2a+3t-1$
$y=2a^2+6t^2-6at-a+3t-1$
$c=2a^2+6t^2-6at-2$
$d=2a^2+6t^2-6at-2a+4t-1$
$a,t$ - what some integers any sign.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/786285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solving a differential equation using the laplace transform involving convolution The problem is the following
The thing that puzzles me here is the integral on the right hand side, so:
How to take the laplace transform on the right hand side?
Any help to get me going would be greatly appreciated.
| The convolution theorem is
\begin{align}
F(s)*G(s) = \int_{0}^{t} g(t-\tau) f(\tau) d\tau
\end{align}
such that
\begin{align}
L\{ f*g\} = L\{ \int_{0}^{t} g(t-\tau) f(\tau) d\tau \} = F(s)G(s).
\end{align}
The differential equation then becomes
\begin{align}
L\{ y^{''} + 3 y^{'} + 2y \} = L\{ H(t-\pi) \sin(2t) \} + \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/790607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Values of $m$ for which $y^2 + 2xy + 2x -my -3$ can be factorised
For what values of $m$, will the expression $y^2 + 2xy + 2x -my -3$ be capable of resolution into two rational linear factor?
This is how I did it:
$$y^2 + 2xy + 2x -my -3 = y^2+(2x-m)y+2x-3$$
This can always be factorized if $b^2-4ac>0$, so if $4ac$ w... | Hint: The factorization can be taken to have shape $(y+ax+b)(y+cx+d)$. Since there is no $x^2$ term, we have $a=0$ or $c=0$. We can take $c=0$. The term in $xy$ is $2xy$, so $a=2$. Continue. We are pretty close to the end.
Added: Your discriminant approach will also work. The discriminant is
$(2x-m)^2-4(2x-3)$. This e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/796117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Reducing Function to be Expressed in Terms of 1 Trigonometric Function I am looking for a way to express $\tan(x) + \sec(x)$ as a function expressed in terms of a single trigonometric function. So far I have it down to:
$$
\frac{\sqrt{1 - \cos^2 x} + 1}{\cos(x)}
$$
Is there a more clean way to define this. Basically I ... | $\tan x + \sec x = \dfrac{\sin x + 1}{\cos x} = \dfrac{(\sin(\frac{x}{2}) + \cos(\frac{x}{2}))^2}{(\cos(\frac{x}{2}) - \sin(\frac{x}{2}))(\cos(\frac{x}{2}) + \sin(\frac{x}{2}))} = \dfrac{\sin(\frac{x}{2}) + \cos(\frac{x}{2})}{\cos(\frac{x}{2}) - \sin(\frac{x}{2})} = \dfrac{\tan(\frac{x}{2}) + 1}{1 - \tan(\frac{x}{2})}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/798208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Formula for the number of integer solutions of an equation (using generating functions) Let $a_n$ be the number of integer solutions of $$i+3j+3k=n$$ where $i \geq 0, j \geq 2, k \geq 3$.
I want to use the generating function of $(a_n)_{n \in \mathbb N}$ to get a formula for $a_n$.
We just introduced generating functio... | OK, use generating functions. Note that $1 - z^3 = (1 - \omega z) (1 - \omega^2 z) (1 - z)$, where $\omega = -\frac{1}{2} + \mathrm{i} \frac{\sqrt{3}}{2}$ and $\omega^3 = 1$:
\begin{align}
[z^n] z^{15} (1 + z &+ z^2 + \ldots) (1 + z^3 + z^6 + \ldots)^2 \\
&= [z^{n - 15}] \frac{1}{(1 - z) (1 - z^3)^2} \\
&= [z^{n - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
What is wrong with this limit $\lim_{x \rightarrow \infty} \frac{2x^4-1}{-4x^5+x^2}$? The arrow indicates an elimination due to numbers becoming negligible at high values of x.
The correct answer is 0, but I got infinity and I'm not sure where my reasoning went wrong.
$$\lim_{x \rightarrow \infty} \frac{2x^4-1}{-4x^5+... | Your mistake is not that you did not evaluate the numerator and denominator at the same time or in the same way. Your mistake is strictly a computational one: $$\lim_{x \rightarrow \infty} \frac{x^4(2-\frac{1}{x^4})}{x^5(-4+\frac{x^2}{x^5})}=\lim_{x \rightarrow \infty} \frac{x^4(2)}{x^5(\frac{1}{x^3})}$$ is incorrect.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Finding integral solutions to the equation $x^4-ax^3-bx^2-cx-d=0$ How many integral solutions exist for the equation:
$$x^4-ax^3-bx^2-cx-d=0\qquad a\ge b\ge c\ge d\qquad a,b,c,d\in\Bbb{N}$$
I have no idea where to begin even.Please help.
| Let's find integer root $x$.
As $d\ne 0$, then $x\ne 0$.
Denote $y=1/x$.
Then equation for $y$:
$$
ay+by^2+cy^3+dy^4=1.\tag{1}
$$
A. If $x=-n$, where $n\in\mathbb{N}$, then $y=-\dfrac{1}{n}$, and $(1) \implies$
$$
\dfrac{-a}{n}+\dfrac{b}{n^2}+\dfrac{-c}{n^3}+\dfrac{d}{n^4}=1.\tag{2}
$$
$$
\dfrac{-an+b}{n^2}+\dfrac{-cn... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Finding generators for an ideal of $\Bbb{Z}[x]$ We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime natural numbers $p$. How can I determine a finite number of elements which gen... | $$ (1 - 11 x - x^2 + 5 x^3 - 5 x^4) (2 x^2 + 2) + 4 (1 + x) (3 x^3 + 3) -
2 (1 - x) (5 x^5 + 5) = 4$$
$$-112 (1 - x^3 - x^5) (5 x^5 + 5) - 80 (x + x^3) (7 x^7 + 7) + \\
187 (1 - x^3 + x^6 - x^9) (3 x^3 + 3) + 51 x (11 x^{11} + 11) = x+1$$
Conversely,
$2x^2+2 = 2(x+1)(x-1)+4$, $px^p+p = p(x+1)(1-x+\dots + x^{p-1})$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Calculation of integral $\int\exp \left(-\alpha \sin^2 \left(\frac{x}{2} \right) \right) dx$ Given $\alpha$ is a constant. How to calculate the following integral?
\begin{equation}
\int \exp \bigg(-\alpha \sin^2 \bigg(\frac{x}{2} \bigg) \bigg) dx
\end{equation}
Thanks for your answer.
| Let $u=\dfrac{x}{2}$ ,
Then $x=2u$
$dx=2~du$
$\therefore\int e^{-\alpha\sin^2\frac{x}{2}}~dx$
$=2\int e^{-\alpha\sin^2u}~du$
$=2\int\sum\limits_{n=0}^\infty\dfrac{(-1)^n\alpha^n\sin^{2n}u}{n!}du$
$=2\int\left(1+\sum\limits_{n=1}^\infty\dfrac{(-1)^n\alpha^n\sin^{2n}u}{n!}\right)du$
For $n$ is any natural number,
$\int\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
How to solve $P=\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^3}\right)\ldots \infty$ How do I find the following product
$$P=\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^2}\right)\left(1+\frac{1}{3^3}\right)\ldots \infty$$
| This is not an answer but I hope others will help to conclude something.
First of all, take a logarithm from both sides:
$$
\log P=\log\prod_n\left(1+\frac{1}{3^n}\right)=\sum_n\log\left(1+\frac{1}{3^n}\right)
$$
Now using Taylor series, you get:
$$
\log(1+\frac{1}{3^m})=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
How prove $\left(\frac{b+c}{a}+2\right)^2+\left(\frac{c}{b}+2\right)^2+\left(\frac{c}{a+b}-1\right)^2\ge 5$ Let $a,b,c\in R$ and $ab\neq 0,a+b\neq 0$. Find the minimum of:
$$\left(\dfrac{b+c}{a}+2\right)^2+\left(\dfrac{c}{b}+2\right)^2+\left(\dfrac{c}{a+b}-1\right)^2\ge 5$$
if and only if $$a=b=1,c=-2$$
My idea: Since ... | Continuing from Calvin Lin' answer, the minimum value will be otained when all partial derivatives will be zero. The only partial derivative which is easy to write is with respect to $c$ which only appears in numerator. Almost as Calvin Lin wrote, this derivative cancels when $$c=-\frac{b (a+b) (2 a+b)}{a^2+a b+b^2}$$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/807404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
How to prove: prove $\frac{1+\tan^2\theta}{1+\cot^2\theta} = \tan^2\theta$ I need to prove that $\frac{1+\tan^2\theta}{1+\cot^2\theta}= \tan^2\theta.$
I know that $1+\tan^2\theta=\sec^2\theta$ and that $1+\cot^2\theta=\csc^2\theta$, making it now $$\frac{\sec^2\theta}{\csc^2\theta,}$$ but I don't know how to get it do... | Look at $\sin^2 \theta + \cos^2 \theta = 1$. Dividing by $\cos^2 \theta$ we get $\tan^2 \theta + 1 = \sec^2 \theta$. Try to find another relation by dividing by $\sin^2 \theta $ and see what appears.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/810453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 9,
"answer_id": 2
} |
Series Question: $\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$ How to compute the following series:
$$\sum_{n=1}^{\infty}\frac{n+1}{2^nn^2}$$
I tried
$$\frac{n+1}{2^nn^2}=\frac{1}{2^nn}+\frac{1}{2^nn^2}$$
The idea is using Riemann zeta function
$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$
but the term $2^n$ makes complicate... | Consider Maclaurin series of natural logarithm
$$
\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}{n}.
$$
Taking $x=\dfrac12$ yields
\begin{align}
\ln\left(1-\frac12\right)&=-\sum_{n=1}^\infty\frac{1}{2^n\ n}\\
\ln2&=\sum_{n=1}^\infty\frac{1}{2^n\ n}.
\end{align}
Now, dividing the Maclaurin series of natural logarithm by $x$ yiel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
Prove the inequality $\frac{1}{1999}<\frac{1\cdot3\cdot5\cdot7\cdots1997}{2\cdot4\cdot6\cdot8\cdots1998}<\frac{1}{44}$ prove this inequality.
$\dfrac{1}{1999}<\dfrac{1\cdot3\cdot5\cdot7\cdots1997}{2\cdot4\cdot6\cdot8\cdots1998}<\dfrac{1}{44}$
I have tried to convert this series in factorial form. I am not getting what ... | Just to expand on the comment I made, let $X = \dfrac{1\cdot3\cdot5\cdot7\cdots1997}{2\cdot4\cdot6\cdot8\cdots1998}$. Then the right inequality is equivalent to showing
$$ \frac1{X^2} > 44^2 \iff \frac{2^2}{1 \cdot 3} \cdot \frac{4^2}{3 \cdot 5}\cdot \frac{6^2}{5 \cdot 7} \cdots \frac{1998^2}{1997 \cdot 1999}> \frac{4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/814005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
$4 \sin 72^\circ \sin 36^\circ = \sqrt 5$ How do I establish this and similar values of trigonometric functions?
$$
4 \sin 72^\circ \sin 36^\circ = \sqrt 5
$$
| A different approach
$ a = n\pi/5$
$5a = n\pi$
$3a = n\pi-2a$
$\sin 3a = \sin{(n\pi-2a)}$
$\sin 3a = \sin2a$
Expand using identities
Remove a sin
Square both sides
You get a quadratic in $\sin^2$
Let $\sin^2 a = x$
So
$16x^2 -20 a +5 = 0$
So
$\sin^2 72 \cdot\sin^2 36 = 5/16$
Hence
$4\sin72\sin36 = \sqrt5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/816195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Solve a system of linear equations $\newcommand{\Sp}{\phantom{0}}$There is a system of linear equations:
\begin{alignat*}{4}
&x - &&y - 2&&z = &&1, \\
2&x + 3&&y - &&z =-&&2.
\end{alignat*}
I create the matrix of the system:
$$
\left[\begin{array}{rrr|r}
1 & -1 & -2 & 1 \\
2 & 3 & -1 & -2
\end{array}\rig... | This is the problem of finding the complete solution to $Ax=b$. To learn how to do this, you may want to watch this video or Gilbert Strang's full video lecture or notes on the topic.
The idea is that we need to find
(a) the null space $N$ of $ \left[\begin{array}{rrr|r}
1 & -1 & -2 \\
2 & 3 & -1
\end{array}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/817183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$
My approach :
I used $\sin A +\sin B = 2\sin(A+B)/2\times\cos(A-B)/2 $
$\Rightarrow \sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7} = 2\sin(3\pi)/7\times\co... | The idea in this answer is: if you can introduce a trigonometric sum with arguments in arithmetic progression, then you have a nice formula that does all the job (see here for the statement and two proofs).
Let
$$S=\sin \frac{2\pi}{7}+\sin \frac{4\pi}{7}+\sin \frac{8\pi}{7}$$
Then
$$S^2=\sin^2 \frac{2\pi}{7}+\sin^2 \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/818749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 0
} |
How to integrate $\int \frac{1}{\sin^4x + \cos^4 x} \,dx$?
How to integrate
$$\int \frac{1}{\sin^4x + \cos^4 x} \,dx$$
I tried the following approach:
$$\int \frac{1}{\sin^4x + \cos^4 x} \,dx = \int \frac{1}{\sin^4x + (1-\sin^2x)^2} \,dx = \int \frac{1}{\sin^4x + 1- 2\sin^2x + \sin^4x} \,dx \\
= \frac{1}{2}\int \... | Convert the exponential powers to multiple angles. From deMoivre's theorem, with $n\in\mathbb{N}$:
$$
\begin{align}
\left( e^{i \theta} \right)^{n} &= e^{i n\theta} \\
\left( \cos \theta + i \sin \theta \right)^{n} &= \cos n\theta + i \sin n\theta
\end{align}
$$
These intermediate formulas may help:
$$
\begin{align}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 1
} |
Trouble with factoring polynomial fractions and understanding wolfram alpha I hope the community can excuse me if I'm making excessive posts, I have a calculus quiz tomorrow and I want to be as prepared as possible.
I'm going through a review problem on wolfram alpha and I have trouble understanding a step. The problem... | Let's write radicals in fractional notation:
$$\frac{2(x - 5)}{3\sqrt[3]{x}} + x^{2/3} = 0\iff \frac{2(x - 5)}{3x^{1/3}} + x^{2/3}=0.$$
Now, it will become obvious to you: $$\require{cancel}\eqalign{\frac{2(x - 5)}{3x^{1/3}} + x^{2/3}
&=\frac{2x - 10}{3x^{1/3}} + x^{2/3}\\ \ \\
&=\frac{2x}{3x^{1/3}} - \frac{10}{3x^{1/3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/820916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of tricky integral I want to evaluate the integral
$$\int _ {b} ^ {\infty} \mathrm{d} x \, \frac{e ^ {x ^ {2} / s} (b^2 + 3 x ^ 2) ^ {2}}{x (x^2 + b^2)}$$,
where $b$ and $s$ are positive real numbers. I thought of writing $x$ as $x = - i y$. Then $y$ would be a pure imaginary number and the integral would be... | As proposed the integral has a issue with large values, ie $x \rightarrow \infty$, since the integral
is unbounded. This can be remedied by replacing $s$ with $- 1/s$. In this view consider the integral
\begin{align}
I = \int_{b}^{\infty} e^{-s x^{2}} \ \frac{(a x^{2} + b^{2})^{2}}{x ( x^{2} + b^{2}) } \ dx.
\end{align... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to calculate the sum of the infinite series Please help me with the sum of the infinite series:
$$
\Large \frac{1+\frac{\pi^4}{5!}+\frac{\pi^8}{9!}+\frac{\pi^{12}}{13!}+...}
{\frac{1}{3!}+\frac{\pi^4}{7!}+\frac{\pi^8}{11!}+\frac{\pi^{12}}{15!}+...}
$$
| Evaluating the numerator first.
Consider the series
$$\sum_{n=0}^{\infty} \frac{x^{4n}}{(4n)!}=1+\frac{x^4}{4!}+\frac{x^8}{8!}+\cdots$$
Also,
$$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\frac{x^8}{8!}-\cdots$$
$$\cosh x=\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\frac{x^8}{8!}+\cdots$$
Add the above two ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Integrate : $\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}dx$
$$\int \frac{x^2}{(x\cos x -\sin x)(x\sin x +\cos x)}\ dx$$
My approach :
Dividing the denominator by $\cos^2x$ we get $\dfrac{x^2\sec^2x }{(x -\tan x)(x\tan x +1)}$ then
$$\int \frac{x^2\sec^2x}{x^2\tan x -x\tan^2x+x-\tan x}\ dx=\int \frac{x^2(1+\t... | $\bf{Another \; Solution::}$ Let $\displaystyle I = \int\frac{x^2}{(x\sin x+\cos x)\cdot (x\cos x-\sin x)}dx$
Now $\displaystyle I = \int\frac{2x^2}{(x^2\sin 2x+2x\cos^2 x-2x\sin^2 x-\sin 2x)}dx$
So $\displaystyle I = \int\frac{2x^2}{(x^2-1)\sin 2x+2x\cos 2x}dx$
Now $\displaystyle (x^2-1)\sin 2x+2x\cos 2x = \sqrt{(x^2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/821862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 5,
"answer_id": 0
} |
Limit $\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}$ $$\lim_{x\rightarrow\infty}1-x+\sqrt{2+2x+x^2}=\lim_{x\rightarrow\infty}1-x+x\sqrt{\frac{2}{x^2}+\frac 2x+1}=\lim_{x\rightarrow\infty}1=1\neq2$$ as Wolfram Alpha state. Where I miss something?
| For convention, let $\lim$ denote $\lim_{x\rightarrow \infty}$. I avoid big-O notation and power series here and perhaps it is clearer this way?
Now, $$\lim [1-x+\sqrt{2+2x+x^2}] \\
= \lim [1 + \frac{2+2x}{x + \sqrt{2+2x+x^2}}] \\
= \lim [1 + \frac{2+2x}{x(1+\sqrt{2/x^2+2/x+1})}] \\
= \lim [1 + \frac{2/x + 2}{1+\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822017",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Math Competition Question Hong Kong I was (as again) doing practise papers, when i saw this question:
Triangle ABC is divided by segments BD, DF and FE into four triangles. E and D lie on CA and F lies on BC. These four small triangles have equal areas. If BF=2DE, find the ratio of AC:BC.
I have completely no idea on h... |
Since $\triangle CEF$ and $\triangle EDF$ have equal area and equal height, you get $CE=ED$. Without loss of generality, we can assume this length to be equal to $1$. I will also assume the height of these two triangles to be equal to $2$ so that all four areas should be equal to $1$, but read below for comments on th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/822631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Expand $(x-7)^2 + x^2 = (x+2)^2$ algebraically $(x-7)^2 + x^2 = (x+2)^2$
$(x-7)(x-7) + x^2 = (x+2)(x+2)$
$x^2 -7x -7x + 49 + x^2 = x^2 + 4x + 4$
$x^2 + 18x - 45 = x^2 + x^2$
From that point on, everything I do is incorrect. I don't know what to do with the three $x^2$.
| You don't need to expand the squares. Use $a^2-b^2 = (a+b)(a-b)$ to get
$$
x^2=(x+2)^2-(x-7)^2=(x+2+x-7)(x+2-x+7)=9(2x-5)=18x-45
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/822717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Integration by parts - $\int \ln (2x+1) \text{dx}$ Use integration by parts to find
$\int \ln (2x+1) \text{dx}$.
So far I have:
$$x\ln(2x+1)-\int\dfrac{2x}{2x+1}dx+c$$
Using integration by substitution to find the integral
$$u=2x+1\Rightarrow\text{du}=2\text{dx}$$
$$\int\dfrac{2x}{2x+1}\cdot\dfrac{1}{2}\text{du}=\int x... | $$
\begin{aligned}
\int \ln (2 x+1) d x &=\frac{1}{2} \int \ln (2 x+1) d(2 x+1) \\
&=\frac{(2 x+1) \ln (2 x+1)}{2}-\frac{1}{2} \int(2 x+1) \frac{2 d x}{2 x+1} \\
&=\frac{(2 x+1) \ln (2 x+1)}{2}-x+C
\end{aligned}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/823162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Find a closed form for $\sum_{k=0}^{n} k^3$ Find a closed form for $\sum_{k=0}^{n} k^3$.
I would appreciate ideas for approaching questions like this in general as well.
Thanks.
| First, we know that: $$\sum_{i \mathop = 1}^n i = \frac {n \left({n + 1}\right)} 2.$$ Thus: $$\left({\sum_{i \mathop = 1}^n i}\right)^2 = \frac{n^2 \left({n + 1}\right)^2} 4.$$ Next we use induction on $n$ to show that $$\sum_{i \mathop = 1}^n i^3 = \frac{n^2 \left({n + 1}\right)^2} 4.$$
The base case holds since $$1^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/823897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
How prove $\frac{1-xy}{\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{(1-x)^2+(1-y)^2}}\le\frac{\sqrt{5}-1}{4}$ Question:
let $x,y\in [0,1]$, show that
$$\dfrac{1-xy}{\sqrt{1+x^2}+\sqrt{1+y^2}+\sqrt{(1-x)^2+(1-y)^2}}\le\dfrac{\sqrt{5}-1}{4}$$
Thank you (I think this inequality can use Geometric interpretation)
my idea:
$$\Longl... | The in equality becomes $\sqrt{1+a^2}+\sqrt{1+b^2}+\sqrt{(1-a)^2+(1-b)^2}\geq (1+\sqrt{5})(1-ab)$
We'll prove that
$\sqrt{(1-a)^2+(1-b)^2}+\sqrt{5}ab\geq \sqrt{1+(1-a-b)^2}$ (1)
$\sqrt{1+a^2}+\sqrt{1+b^2}+ab\geq 1+\sqrt{1+(a+b)^2}$ (2)
To prove (1) you just need to square it
To prove (2), set :
A=$1+\sqrt{1+(a+b)^2}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/827674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges. I must prove, that sum diverges, but...
$$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$
$$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$
$$\lim_{n \to \infty} \frac{ \sqrt[n]{\arctan{n}} }{ \sqrt[n]{n} } = \frac{1}{1} = 1$$
Cauchy's co... | At the end, you have to take the limit as $n$ and $p$ go to infinity, or, if you prefer, the inequality $|S_{n+p}-S_n|\lt \varepsilon$ has to take place for any $n\geqslant N(\varepsilon)$ and $p\geqslant 0$.
Instead, use the inequality $\arctan n\geqslant \pi /4$ for $n\geqslant 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/827854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Intersection of ellipse and hyperbola at a right angle Need to show that two functions intersect at a right angle.
Show that the ellipse
$$ \frac{x^2}{a^2} +\frac{y^2}{b^2} = 1
$$
and the hyperbola
$$
\frac{x^2}{α^2} −\frac{y^2}{β^2} = 1
$$
will intersect at a right angle if
$$α^2 ≤ a^2 \quad \text{and}\quad a^2 − b^2... | Note that if $a^2=b^2+\alpha^2+\beta^2$ holds, then $a^2\ge \alpha^2$ holds since $$a^2=(b^2+\beta^2)+\alpha^2\ge 0+\alpha^2=\alpha^2.$$
There are four intersection points $(x,y)$ where
$$x^2=\frac{a^2\alpha^2 (b^2+\beta^2)}{\alpha^2 b^2+a^2\beta^2},\qquad y^2=\frac{\beta^2b^2(a^2-\alpha^2)}{\alpha^2b^2+a^2\beta^2}=\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/829621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
} |
Proof by induction that $(1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2$ I'm sitting with the proof in front of me, but I do not understand it.
$$A = \{n \in Z^{++} \mid (1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2\}$$
The first step of proof by induction is simple enough,to prove that $1 \in A$
$1^3 = ... | The last step uses the identity:
$$(a + b)^2 = a ^2 + 2ab + b^2$$
So:
$$(\underbrace{1 + 2 + \cdots + n}_a + \underbrace{(n + 1)}_b)^2 =\\= (1 + 2 + \cdots + n)^2 + 2(1 + 2 + \cdots + n)(n+1) + (n + 1)^2\tag{1}$$
By induction hypotesis we have that $(1 + 2 + \cdots + n) ^2 = (1^3 + 2^3 + \cdots + n^3)$. We also use an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/832756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Partial fraction (doubt) I have this partial fraction
$$\displaystyle\frac{1}{(2+x)^2(4+x)^2}$$
I tried to resolve using this method:
$$\displaystyle\frac{A}{2+x}+\displaystyle\frac{B}{(2+x)^2}+\displaystyle\frac{C}{4+x}+\displaystyle\frac{D}{(4+x)^2}$$
$$1=A(2+x)(4+x)^2+B(4+x)^2+C(4+x)(2+x)^2+D(2+x)^2$$
When x=-2
$$1=... | Try another value of $x$ as well, say $x=2$, giving $1 = 144A + 9 + 64C + 4$, or $144A + 96C = -12$. Combined with the equation you already have, $32A+16C=-4$, solving gives $A=-\frac{1}{4}$, $C=\frac{1}{4}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/833170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Factor the Expression completely$ (a+b)^2 - (a-b)^2$ I don't understand this question. The answer in the book is $4ab$, but how is that term a factor?
I was thinking along the line that this was a difference of squares example. $a^2-b^2 = (a+b)(a-b)$
My answer is $[(a+b)-(a-b)][(a+b)+(a-b)]$
What do I not understand? ... | Just See this,
$(a+b)^2 = a^2 + b^2 +2ab$,
$(a-b)^2 = a^2 + b^2 -2ab$
Subtract them,
You will get $4ab$
Ok I will show that , you have already done everything
$(a+b)^2 - (a-b)^2 = [(a+b)-(a-b)][(a+b)+(a-b)]$
$\hspace{2cm} = [a+b-a+b][a+b+a-b]$
$\hspace{2cm} = [a-a+b+b][a+a+b-b]$
$\hspace{2cm} = [2b][2a]$
$\hspace{2cm}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/834111",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Solve trigonometric equation $\sin14x - \sin12x + 8\sin x - \cos13x= 4$ I am trying to solve the trigonometric equation
$$ \sin14x - \sin12x + 8\sin x - \cos13x= 4 $$
The exact task is to find the number of real solutions for this equation on the range $[0, 2\pi]$. Thanks.
| We can use $\sin{x}-\sin{y}=2\sin{\frac{x-y}{2}}\cos{\frac{x+y}{2}}$.
So, $$(\sin{14x}−\sin{12x})+8\sin{x}−\cos13x=4 \Longleftrightarrow 2\sin{x}\cos{13x}−\cos13x+8\sin{x}-4=0$$ $$\Longleftrightarrow 2\cos{13x}(\sin{x}-\frac{1}{2})+8(\sin{x}-\frac{1}{2})=0 \Longleftrightarrow (\sin{x}-\frac{1}{2})(2\cos{13x}+8)=0$$
So,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/837593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove that diophantine equation has only two solutions. I am looking at the following exercise:
$$\text{Prove that the diophantine equation } x^4-2y^2=1 \text{ has only two solutions.}$$
That's what I thought:
We could set $x^2=k$,then we would have $k^2-2y^2=1 \Rightarrow 2y^2=k^2-1 \Rightarrow 2 \mid k^2-1$
Can I use... | You are doing it right. $x^4 - 2 y^2 = 1 \Leftrightarrow x^4 -1 = 2 y^2 \Leftrightarrow (x-1 ) ( x+ 1 ) ( x^2 +1 ) = 2 y^2$.
Now prove that $x$ is odd, then $x = 2 k +1 $, and show that $k ( k+1) ( 2 k^2 + 2 k + 1)$ is a square, and since each factor is relatively prime, $k$ and $k+1$ are squares, conclude that $k=0$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/839586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to sum $\frac{1}{9} + \frac{1}{18}+\frac{1}{30}+\frac{1}{45} + ......$ How to sum this series :
$\frac{1}{9} + \frac{1}{18}+\frac{1}{30}+\frac{1}{45} + \frac{1}{65}......$
I am not getting any clue only a hint will be suffice please help. thanks..
| Observe
\begin{align*}
\frac{1}{9}+\frac{1}{18}+\frac{1}{30}+\frac{1}{45}+\ldots&=\frac{1}{3+3\cdot2}+\frac{1}{3+3\cdot2+3\cdot3}+\frac{1}{3+3\cdot2+3\cdot3+3\cdot4}+\ldots\\
&=\sum_{k=2}^{\infty}{\frac{1}{3\sum_{j=1}^{k}{j}}}\\
&=\sum_{k=2}^{\infty}{\frac{1}{3\left[\frac{k(k+1)}{2}\right]}}\\
&=\sum_{k=2}^{\infty}{\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/840270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
$a^{12} \equiv 1 \pmod{35}$,knowing that $(a,35)=1$ Prove that $\forall a \text{ with } (a,35)=1:$
$$a^{12} \equiv 1 \pmod{35}$$
$$35 \mid a^{12}-1 \Leftrightarrow 5 \cdot 7 \mid a^{12}-1 \overset{(5,7=1)}{ \Leftrightarrow} 5 \mid a^{12}-1 \text{ and } 7 \mid a^{12}-1$$
Therefore, $\displaystyle{ a^{12} \equiv 1 \pmod{... | Hint : For $(a,5)=1$ only possibilities are : $a=5k+1;5k+2;5k+3;5k+4$
$(5k+1)^{12}\equiv 1\mod5$ case should be clear...
$(5k+2)^{12}\equiv 1\mod 5$ should be clear if you know what $2^{12}$ is...
$(5k+3)^{12}\equiv 1\mod 5$ should be clear if you know what $3^{12}$ is...
$(5k+4)^{12}\equiv 1\mod 5$ should be clear i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/842741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Trigonometric equation with complex numbers Let $x$, $y$, and $z$ be real numbers such that $\cos x+\cos y+\cos z=\sin x+\sin y+\sin z=0$.
Prove that $\cos 2x+\cos 2y+\cos 2z=\sin 2x+\sin 2y+\sin 2z=0$.
Starting with the given equation, I got that $i\sin x+i\sin y+i\sin z=0$.
Adding this to the other part of the given... | Multiply the equations by $\sin(x+y+z)$ and $\cos(x+y+z)$ and subtract them to get
$$\sin(x+y+z)\cos x -\cos(x+y+z)\sin x +\\
\sin(x+y+z)\cos y -\cos(x+y+z)\sin y +\\
\sin(x+y+z)\cos z -\cos(x+y+z)\sin z\\
=\sin(y+z)+\sin(x+z)+\sin(x+y)=0$$
In a similar way we get
$$\cos(y+z)+\cos(x+z)+\cos(x+y)=0$$
Now if we squa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843022",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Simplest form of $h'(y)$ given $h(y)= (1-3y^2)^5 \cdot ( y^2 + 2)^6$ Find $h'(y)$ in the simplest form if the $$h(y)= (1-3y^2)^5 \cdot ( y^2 + 2)^6$$
My answer was: $$-30y(1-3y^2)^4 \cdot (y^2+2)^6 + 12y(y^2+2)^5 \cdot (1-3y^2)^5$$
But according to wolfram alpha the answer was $$ -30y(1-3y^2)^4 \cdot (y^2+2)^6 $$ onl... | You differentiated correctly.
Since you have been asked to write the derivative in its simplest form:
you can factor out common factors in each term of your sum:
$$-30y(1-3y^2)^4 \cdot (y^2+2)^6 + 12y(y^2+2)^5(1-3y^2)^5$$ $$ = 6y(1-3y^2)^4(y^2 +2)^5\Big((-5)(y^2+2) + 2(1-3y^2)\Big)$$ $$ = 6y(1-3y^2)^4(y^2 +2)^5\Big(-5... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/843855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Locus of vertex of triangle moving inside circle A right triangle with sides $3,4$ and $5$ lies inside the circle $2x^2+2y^2=25$. The triangle is moved inside the circle in such a way that its hypotenuse always forms a chord of the circle. The locus of the vertex opposite to the hypotenuse is ____ ?
A) $2x^2+2y^2=1$
B)... | Some of the results are summarized into the following figure.
$θ = \angle DAX - \angle CAX = sin^{-1} \dfrac {2.5}{2.5√2} – sin^{-1} \dfrac {2.4}{4} = 8.130xxxxxx^0$
In ⊿CAD, $r^2$, the square of the required radius = … by cosine law … = 0.5
Thus, the required equation of the circle is $2x^2 + 2y^2 = 1$
Note:- The st... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/844820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding $ \lim_{x \to - \infty} \left( \frac{1 - x^{3}}{x^{2} + 7 x} \right)^{5} $. I have to evaluate the following limit:
$$
\lim_{x \to - \infty} \left( \frac{1 - x^{3}}{x^{2} + 7 x} \right)^{5}.
$$
What I did was to divide both the numerator and the denominator of the expression inside the parentheses by $ \dfrac{1... | Consider dividing top and bottom of $\dfrac{1-x^3}{x^2+7x}$ by $x^2$. We get
$$\frac{\frac{1}{x^2}-x}{1+\frac{7}{x}}.$$
Now let $x$ get large negative. It is clear that the bottom approaches $1$, and that the top blows up.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/845243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find all integer solution Find all integer solutions such that $$a+1|2a^2+9$$
Solution. I could solve this by writing $$\frac{2a^2+9}{a+1}=2a-2+\frac{11}{a+1}.$$ So, the only integer solution for the last equation are $a=10, a=-12.$
But, i want to get a solution using divisibility properties.
| You pretty much have the right idea. Observe that $a+1 | 2(a^2-1)$. Moreover we can write
$$2a^2+9=2(a^2-1)+11.$$
Therefore for $a+1$ to divide $2a^2+9$, it has to divide $11$ (because $11$ is the difference of two expression both of which are divisible by $a+1$).
Thus the only values of $a+1$ are all divisors of $11$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Is $\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$ a simple extension? Is the extension $$\mathbb{Q}\left( \sqrt[3]{2}, \frac{-1 + i\sqrt{3}}{2}\right):\mathbb{Q}$$ simple? If so find the minimal polynomial and the basis for the extension.
| This extension can be written as $\mathbb{Q}(\sqrt[3]{2}, \zeta_3)$, where $\zeta_3 = e^{2\pi i /3}$. This is a degree 6 extension.
$$p(x) = x^6+3 x^5+6 x^4+3 x^3+9 x+9$$ is the minimal polynomial for $\sqrt[3]{2}+ \zeta_3$. Since this polynomial is degree 6, its roots form number fields of degree 6 over $\mathbb{Q}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/846471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Math probability combination explanation A group of four components is known to contain two defectives. An inspector tests the components one at a time until the two defectives are located. once she locates the two defectives, she stops testing, but the second defective is tested to ensure accuracy. let Y denote the nu... | The numbers are small, so we do not need general theory to do the calculation.
The random variable $Y$ can take on values $2$, $3$, or $4$.
$\Pr(Y=2)$: We need to have the first defective, and the second. The probability the first is defective is $\frac{2}{4}$. Given the first was defective, there are $3$ items left, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/847836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
expected value and variance of the difference of number of people in a row. I need to calculate the expected value and the variance of the following variable:
$n$ people sit in a row, among them person 'a' and person 'b'.
Define $X$ to be the amount of people between 'a' and 'b'. Calculate $E(X)$ and $Var(X)$.
I have... | $P\left(X=k\right) = \frac{\mbox{#positions of A and B with $k$ seats between them}}{\mbox{#total positions for A and B}}$.
When A is to the left of B, there are $n-k-1$ possible positions they can have with $k$ seats between them. When A is to the right of B, this is a mirror image so there is a further $n-k-1$ positi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/848157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
L'Hôpital's Rule and Infinite Limits I was wondering if anyone could help me with computing a limit using L'Hôpital's Rule.
Using L'Hôpital Rule for the following limit, I get the following result:
\begin{equation}
\lim_{x \to 0} \frac{e^x-1-x^2}{x^4+x^3+x^2} \therefore \lim_{x \to 0} \frac{e^x-1-x^2}{x^4+x^3+x^2} \sta... | The right way to proceed is the following from what you have found
$$\lim_{x \to 0^+} \frac{e^x-2x}{4x^3+3x^2+2x} = \infty \implies \lim_{x \to 0^+} \frac{e^x-1-x^2}{x^4+x^3+x^2} = \infty$$
$$\lim_{x \to 0^-} \frac{e^x-2x}{4x^3+3x^2+2x} = -\infty \implies \lim_{x \to 0^-} \frac{e^x-1-x^2}{x^4+x^3+x^2} = -\infty$$
where... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/848673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Evaluation of $ \int \tan x\cdot \sqrt{1+\sin x}dx$ Calculation of $\displaystyle \int \tan x\cdot \sqrt{1+\sin x}dx$
$\bf{My\; Try::}$ Let $\displaystyle (1+\sin x)= t^2\;,$ Then $\displaystyle \cos xdx = 2tdt\Rightarrow dx = \frac{2t}{\sqrt{2-t^2}}dt$
So Integral is $\displaystyle = \displaystyle 2\int \frac{t^2}{\s... | I think you made some mistakes is the substitution,
$$t^2=\sin x +1$$ so
$$2tdt=\cos x dx$$
now $$\frac{\tan x}{\cos x}= \frac{\sin x}{\cos^2 x}= \frac{\sin x}{1-\sin^2 x}=
\frac{t^2-1}{1-(t^2-1)^2 }=\frac{t^2-1}{2t^2-t^4 }$$
So $$\int \tan x \sqrt{\sin x +1} dx= \int \frac{\tan x}{\cos x} \sqrt{\sin x +1} \cos x dx=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/849580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 1
} |
How to calculate $\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}\ dx$? How to calculate $$\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}dx?$$ I really don't know how to attack this integral. I tried $u=x^2 + 4x + 5$ but failed miserably. Help please.
| \begin{align}
\int\frac1{x+1+\sqrt{x^2+4x+5}}\ dx&=\int\frac1{x+1+\sqrt{(x+2)^2+1}}\ dx\\
&\stackrel{\color{red}{[1]}}=\int\frac{\sec^2y}{\sec y+\tan y-1}\ dy\\
&\stackrel{\color{red}{[2]}}=\int\frac{\sec y}{\sin y-\cos y+1}\ dy\\
&\stackrel{\color{red}{[3]}}=\int\frac{1+t^2}{t(1+t)(1-t^2)}\ dt\\
&\stackrel{\color{red}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/850500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
How to calculate the integral $\int_{-1}^{1}\frac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$? The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
| Rearrange the integral
$$\int \frac{1}{1+z}\sqrt[3]{\frac{1+z}{1-z}}dz$$
Now set $$t^3=\frac{1+z}{1-z}$$
so
$$z=\frac{t^3-1}{t^3+1}=1-\frac{2}{t^3+1}$$
$$dz=\frac{6t^2}{(t^3+1)^2}dt$$
And the integral becomes
$$\int \frac{1}{1+z}\sqrt[3]{\frac{1+z}{1-z}}dz=\int \frac{t^3+1}{2t^3}t\frac{6t^2}{(t^3+1)^2}dt=\int \frac{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Find the minimum of $\displaystyle \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$ Is it possible to find the minimum value of $E$ where
$$E = \frac{1}{\sin^2(\angle A)} + \frac{1}{\sin^2(\angle B)} + \frac{1}{\sin^2(\angle C)}$$for any $\triangle ABC$.
I've got the feeling that $\... | fix $A$, and now we have:
$$\frac{1}{\sin^2(B)}+\frac{1}{\sin^2(C)}\geq \frac{2}{\sin^2\left(\frac{B+C}{2}\right)}$$
becaues $\dfrac{1}{\sin^2(x)}$ is a convex function over $[0,\pi]$. thus when we fix an angle like $A$ it is better that two other angles be equal. Now the function $\dfrac{1}{\sin^2(A)}+\dfrac{1}{\sin^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/851285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Find the exact value of $\sin (\theta)$ and $\cos (\theta)$ when $\tan (\theta)=\frac{12}{5}$ So I've been asked to find $\sin(\theta)$ and $\cos(\theta)$ when $\tan(\theta)=\cfrac{12}{5}$; my question is if $\tan (\theta)=\cfrac{\sin (\theta) }{\cos (\theta)}$ does this mean that because $\tan (\theta)=\cfrac{12}{5}$... | The error is that $\tan\theta$ merely gives the ratio of $\sin\theta$ and $\cos\theta$. If it is given as a fraction, that does not mean that the numerator must be $\sin\theta$ and the denominator must be $\cos\theta$. It is necessary to use trigonometric identities to solve for $\sin\theta$ and $\cos\theta$.
We have
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/852337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 5
} |
How to solve this linear equation? which has an x on each side I have made this equation.
$5x + 8 = 10x + \dfrac{3}{6}$
And I have achieved this result:
$x = 9$
Is my result correct?
I have already posted two other questions related to this topic, I'm a programmer and am learning Math out of my interest, this is not h... | $5x + 8 = 10x + 3/6$
Let's gather all the x's on one side and the other stuff on the other side as well as reduce the fraction to lowest terms first:
$10x-5x = 8-\frac{1}{2}$
$5x = 8-.5$
$5x = 7.5$
$x = 1.5$ or $x=\frac{3}{2}$ or $x=1+\frac{1}{2}$
Now, to compute each side as a verfication:
$LHS= 5x+8 = 5*\frac{3}{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/853062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
} |
The sum of three consecutive cubes numbers produces 9 multiple I want to prove that $n^3 + (n+1)^3 + (n+2)^3$ is always a $9$ multiple
I used induction by the way.
I reach this expression: $(n+1)^3 + (n+2)^3 + (n+3)^3$
But is a lot of time to calculate each three terms, so could you help me to achieve the induction fo... | As an alternative to induction, we take any $3$ consecutive cubes as follows:
$$(n-1)^3 + n^3 + (n+1)^3$$
$$= 3n^3 + 6n$$
$$=3n(n^2 +2)$$
Notice that
$$\begin{align}n(n^2 + 2) &\equiv n(n^2-1)\pmod 3 \\&\equiv (n-1)(n)(n+1)\pmod 3 \end{align}$$
Since either one of $(n-1),n$ or $(n+1)$ must be divisible by $3$, it fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/859572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 5
} |
Evaluation of $\int\sqrt[4]{\tan x}dx$ Evaluation of $\displaystyle \int\sqrt[4]{\tan x}dx$
$\bf{My\; Try}::$ Let $\tan x = t^4\;\;,$ Then $\sec^2 xdx = 4t^3dt$. So $\displaystyle dx = \frac{4t^3}{1+t^8}dt$
So Integral Convert into $\displaystyle 4\int\frac{t^4}{1+t^8}dt = 2\int \frac{(t^4+1)+(t^4-1)}{t^8+1}dt$
So Inte... | Transforming the integral into an integral of rational function by letting $y^{4}=\tan x$, then
$$
I=4 \int \frac{y^{4}}{1+y^{8}} d y
$$
By my post,
$$ \displaystyle \int \frac{x^{4}}{1+x^{8}} d x =\frac{1}{4 \sqrt{2}}\left[\frac{1}{2 \sqrt{2+\sqrt{2}}} \ln \left|\frac{x^{2}-\sqrt{2+\sqrt{2}} x+1}{x^{2}+\sqrt{2+\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/860315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Reliability Probability problem What is the Probability that at least one close path is formed from $A$ to $B$ where each switch has a probability of close $p$ and each switch is mutually independent of each other?
Proposed Solution
Let event $A$ be such that
$$\begin{align}
A &= \{\text{Current Flows From A to B}\... | Notice that
$$\begin{align}
p[A] &= p^4 + p^2(1-p)^2 + p^3(1-p) + p^3(1-p) + p^3(1-p) \\
&= p^4+3p^3(1-p)+p^2(1-p)^2\\
&= -p^4+p^3+p^2
\end{align}$$
where $q = 1-p$ and $-p^4+p^3+p^2=p^2(1+pq)$ so as others nocited: your calculation for the event probability is correct.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/860522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Convergence of an improper integral $\int_0^\infty \frac {1-e^{-x}} {\sqrt x^3}dx$ I need to find if
$$\int_0^\infty \frac {e^{-x}-1} {\sqrt x^3}dx$$
converges.
Let $$f(x) = \int_0^\infty \frac {1- e^{-x}} {\sqrt x^3}dx \ge 0$$
$$\int_0^\infty \frac {1-e^{-x}} {\sqrt x^3}dx = \int_0^1 \frac {1- e^{-x}} {\sqrt x^3}dx +... | Given
$$
\int_0^\infty \frac{ \exp(-x) - 1 }{ \sqrt{x}^3 } dx.
$$
Use substitution $x = y^2$, then we obtain
$$
\begin{eqnarray}
\int_0^\infty \frac{ \exp(-x) - 1 }{ \sqrt{x}^3 } dx &=& 2 \int_0^\infty \frac{ \exp(-y^2) - 1 }{ y^2 } dy\\
&=& \color{red}{\left[ - 2 \frac{ \exp(-y^2) - 1 }{y} \right]_0^\infty}
- \color{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/863694",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Series Convergence/Divergence $\frac{n^n}{(n+1)^{n+1}}$ Trying to establish whether $\sum x_n$ for $x_n := \frac{n^n}{(n+1)^{n+1}}$ converges or diverges. Here's what I've done so far:
1) n-th term: $x_n < \frac{n^n}{n^{n+1}} = \frac{1}{n}$, so $\lim(x_n) = 0$; n-th term test inconclusive.
2) quotient test: $$
\begin{... | It's not hard to see that $(1+\frac{1}{n})^{-n}> \frac{1}{e}$, so $(1+\frac{1}{n})^{-n}\frac{1}{n+1}>\frac{1}{e}\frac{1}{n+1}$. But $\sum \frac{1}{e}\frac{1}{n+1}$ diverges so the original series diverges too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/864278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What is the closed form for $\sum_{n=1}^\infty \frac1n - \frac1{n+1/p}$? A while ago, I started to look at expressions of the following form:
$$
S_p:=\sum_{n=1}^\infty \frac1n - \frac1{n+1/p},
$$
where $p$ is prime, because otherwise things get too complicated for me at the moment.
What I found so far is the followin... | First, for any $z \in \mathbb{C}$ with $|z| < 1$, we have following expansion
$$-\log(1-z) = \sum_{n=1}^\infty \frac{z^n}{n}$$
Let $\displaystyle\;\omega = e^{\frac{i2\pi}{p}}\;$ be the primitive $p$-root of unity, we know for any integer $n$,
$$\frac{1}{p}\sum_{\ell=0}^{p-1} (\omega^\ell)^n = \delta_p(n)
\stackrel{de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/864609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Diagonalizable Matrix real values Let $A=\begin{pmatrix}
a & -b \\
b & a \\
\end{pmatrix}$. For which real values of $a$ and $b$ is $A$ a diagonalizable matrix?
| Although David's answer already provides a sufficient algebraic explanation, I'd like to give a bit of a geometric one. Consider where this matrix sends the standard orthonormal basis
$$ \begin{pmatrix} a &- b\\ b & a \end{pmatrix}\begin{pmatrix} 1\\ 0\end{pmatrix} = \begin{pmatrix} a\\ b \end{pmatrix} =v_1, \begin{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)
Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer.
I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach useful conclusions.
| For the original version, which asked abuot
$$2014^{2015} + 2015^{2016} + 2016^{2014} + 1$$
then no: This is an even number (sum of two evens and two odds), so not prime. It's also divisible by $3$.
For the new question: The number is divisible by $7$, so it is not prime. This can be computed by hand by reducing every... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/866619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form? A friend shared with me the following puzzle he encountered in a Chinese math competition:
In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, or on either dia... | Part 1.
Prove that it is impossible to place $10$ tokens.
Let's try to build with $10$ tokens.
First, note that in each column/row must be at least $1$ token.
Lets call column with token in $1$st cell as column $a$; and row with token in $2$nd cell as column $b$.
Column $a$ and column $b$ must have at least $2$ tokens.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/867462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Multidimensional integral involving delta functions The question is to compute the following multidimensional integral:
\begin{equation}
\omega^{(T)}({\bf c}) := \int\limits_{{\mathbb R}^{2 T}}
\delta\left( c_{1,1} - \sum\limits_{j=1}^T x_{1,j}^2 \right)
\delta\left( c_{2,2} - \sum\limits_{j=1}^T x_{2,j}^2 \right)
\de... | Below is the result for $T=2$. We have:
\begin{eqnarray}
\int\limits_{{\mathcal R}^4}
\delta\left(c_{1,1} - x_{1,1}^2 - x_{1,2}^2\right)
\delta\left(c_{2,2} - x_{2,1}^2 - x_{2,2}^2\right)
\delta\left(c_{1,2} - x_{1,1}x_{2,1} - x_{1,2}x_{2,2}\right) d x_{1,1} d x_{1,2} d x_{2,1} d x_{2,2} = \\
\int\limits_{{\mathcal R}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
How find the value of the $x+y$ Question:
let $x,y\in \Bbb R $, and such
$$\begin{cases}
3x^3+4y^3=7\\
4x^4+3y^4=16
\end{cases}$$
Find the $x+y$
This problem is from china some BBS
My idea: since
$$(3x^3+4y^3)(4x^4+3y^4)=12(x^7+y^7)+x^3y^3(9y+16x)=112$$
$$(3x^3+4y^3)^2+(4x^4+3y^4)^2=9(x^6+y^8)+16(y^6+x^8)+24x^3y^3(... | Assuming the question is typed correctly as shown, there are two unique real solutions. Let $$\begin{align*} u(z) &= -983749-111132z^3+786432z^4+71442z^6-196608z^8-20412z^9+18571z^{12}, \\ v(z) &= -178112-351232z^3+186624z^4+301056z^6-34992z^8-114688z^9+18571z^{12}. \end{align*}$$ These polynomials have exactly two di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Area of a triangle whose each side is less than 2 and greater than1. What is the area of a triangle if each of its sides is greater than 1 and less than 2?
My Try:Let a,b,c be the sides of triangle,then
Area$=\sqrt{s(s-a)(s-b)(s-c)}=\sqrt{\frac{a+b+c}{2}\frac{b+c-a}{2}\frac{a+c-b}{2}\frac{a+b-c}{2}}=\frac{1}{4}\sqrt{(a... | You can get arbitrarily small area, since you can get arbitrarily close to the degenerate triangle $a=b=1$, $c=2$, which has zero area.
The maximal possible area is obtained when forming equilateral triangle with $a=b=c=2$. (I.e., the area is $\sqrt3$.)
One of possible arguments is the following:
If two sides are shor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/868706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$ $$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$
My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I think they should.
I've also tried using ordinary ... | Choose $t=\tan x$ and then $dt=(1+\tan^2x)dx$.
So your integral is:
$$\int \frac{t}{t^3+3 t^2+2 t+6} dt=\int \frac{t}{(t+3)(t^2+2)} dt.$$
We have:
$$\frac{1}{(t+3)(t^2+2)}=\frac{At+B}{t^2+2}+\frac{C}{t+3}=\frac{(A+C)t^2+(3A+B)t+(3B+2C)}{(t+3)(t^2+2)}.$$
We easily get: $A=-C=-\dfrac{1}{11}$ and $B=\dfrac{3}{11}$.
There... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/869142",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Calculating the area For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them;
Attempt to solve: Limits of the integral are $1$ and $-3$, so I took the definite integral of the diffrence between $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x... | For $-3<x<1$:
$$\frac{x^3+2x^2-8x+6}{x+4} < \frac{x^3+x^2-10x+9}{x+4}$$
So you have to calculate the integral of the difference $\frac{x^3+x^2-10x+9}{x+4}-\frac{x^3+2x^2-8x+6}{x+4}$ to find the area.
EDIT:
$$\frac{x^3+x^2-10x+9}{x+4}-\frac{x^3+2x^2-8x+6}{x+4}=\frac{-x^2-2x+3}{x+4}=-\frac{x^2+4x-2x}{x+4}+\frac{3}{x+4}=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/871646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
find a polynomial whose roots are inverse of squares of roots of $x^3+px+q$ Question is :
Given a polynomial $f(x)=x^3+px+q\in \mathbb{Q}[x]$ find a polynomial whose roots are inverse of sqares of roots of $f(x)$
Supposing $a,b,c$ as roots of $f(x)$ we have :
*
*$a+b+c=0$
*$ab+bc+ca=p$
*$abc=-q$
Now i need to kn... | This has been answered by Mr.Michael But then it is not complete answer..
So, Just for my convenience I would write it explicitly...
Given a polynomial $f(x)\in \mathbb{Q}[x]$ with roots $a,b,c$ we need to find a poly. whose roots are $\dfrac{1}{a^2},\dfrac{1}{b^2},\dfrac{1}{c^2}$
Given that $f(x)=x^3+px+q$ with roo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Show that this expression is a perfect square? Show that this expression is a perfect square?
$(b^2 + 3a^2 )^2 - 4 ab*(2b^2 - ab - 6a^2)$
| Observe that $4ab(2b^2-6a^2)=8ab(b^2-3a^2)$
Using $(A+B)^2=(A-B)^2+4AB$
$$(b^2+3a^2)^2=(b^2-3a^2)^2+4(b^2)(3a^2)$$
$$\implies(b^2+3a^2)^2-8ab(b^2-3a^2)+4a^2b^2$$
$$=(b^2-3a^2)^2+12a^2b^2-8ab(b^2-3a^2)+4a^2b^2$$
$$=(\underbrace{b^2-3a^2}_p)^2 -2(\underbrace{b^2-3a^2}_p)(\underbrace{4ab}_q)+(\underbrace{4ab}_q)^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/872331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Which contour is best for $\int_0^\infty\frac{1}{x^2 + x + 1}dx$ The following is a complex analysis problem. Does anyone have any idea what contour would be good to use?
$$\int_0^\infty\frac{1}{x^2 + x + 1}dx$$
Its roots on the bottom are are $\frac{-1 \pm i\sqrt{3}}{2}$.
| Let $ \displaystyle f(z) = \frac{\log z}{z^{2}+z+1}$ and integrate around a keyhole contour where the branch cut for $\log z$ is placed on the positive real axis.
As the radius of the little circle goes to $0$ and the radius of the big circle goes to $\infty$, $ \int f(z) \ dz$ will vanish along both circles. You can u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/872742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Functional equation with cyclic function. Find all functions $f:\mathbb R \to \mathbb R$ that satisfy:
$$ f(x) + 3 f\left( \frac {x-1}{x} \right) = 7x.$$
Some progress: I plugged-in $\dfrac{x-1}{x}$ and $\dfrac{1}{1-x}$, got a system of equations and solving I got $f(x) = \dfrac{x^3-4x^2-3x-3}{4x^2-4x}$. But after te... | Set $T(x) = \frac{x}{x-1}$. Then the composition $T^3(x) = x$. We have $f(Tx) = a x + b f(x)$ for $ a= \frac{7}{3}$ and $b = -\frac{1}{3}$, so
\begin{align*}
f(x)
&= f(T^3(x)) \\
&= a T^2(x) + b f(T^2 x) \\
&= a T^2(x) + a b T(x) + b^2 f(Tx) \\
&= a T^2(x) + a b T(x) +a b^2 x + b^3 f(x).
\end{align*}
That relation give... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/875030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx$
Compute the indefinite integral
$$
\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx
$$
My Attempt:
$$
\begin{align}
\int\frac{\sqrt{\cos 2x}}{\sin x}\,dx &= \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx\\
&= \int\frac{2\cos^2 x-1}{(1-\cos^2 x)\sqrt{2\cos^2 x-1} }\sin ... | \begin{align}
\int\frac{\sqrt{\cos 2x}}{\sin x}\ dx&=\int\frac{\sqrt{\cos^2x-\sin^2x}}{\sin x}\ dx\\
&\stackrel{\color{red}{[1]}}=\int\frac{\sqrt{t^4-6t^2+1}}{t^3+t}\ dt\\
&\stackrel{\color{red}{[2]}}=\frac12\int\frac{\sqrt{u^2-6u+1}}{u^2+u}\ du\\
&\stackrel{\color{red}{[3]}}=\int\frac{(y^2-6y+1)^2}{(y-1)(y-3)(y+1)(y^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/876175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 5,
"answer_id": 3
} |
If $(a, b,c)$ is a Pythagorean triple with $b, c$ consecutive integers then $c \mid a^b – 1$, proof/disproof? If $(a, b,c)$ is a Pythagorean triple with $b, c$ consecutive integers then $c \mid a^b – 1$, proof/disproof?
Here are some examples:
$(3, 4, 5)$ is a Primitive Pythagorean Triple (PPT), $3^2 + 4^2 = 5^2$, wher... | If $b=c-1$ is divisible by $4$, this is true.
$$a^2+(c-1)^2=c^2 \iff a^2=2c-1.$$
Write $b=4k$, and the above yields: $$a^b = (a^2)^{2k} =(2c-1)^{2k} \equiv 1 (\mod c).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/876344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Simplifying $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}$ In trying to simplify my answer to a problem posted recently, I am trying to show that
$\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}=8\binom{24}{4}$.
I know that $\displaystyle\sum_{k=0}^{20}\binom{23-k}{3}=\binom{24}{4}$, but how can I simplify $\displ... | Maple gives
$$\sum_{k=1}^n k\binom{m+n-k}{m}
=\frac{(m+n)(m+n+1)}{(m+1)(m+2)}\binom{m+n-1}{m}\ ,$$
which can be checked by induction if you wish. So your sum becomes
$$4\binom{24}{4}+\frac{23\times24}{4\times5}\binom{22}{3}
=4\binom{24}{4}
+\frac{23\times24}{4\times5}\frac{4\times20}{23\times24}\binom{24}{4}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/882127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How does $x^3 - \sin^3 x$ become $x^3 + \frac{1}{4}\sin{3x}-\frac{3}{4}\sin x$? I was going through answers on this question and came across this answer and I was wondering how the user arrived at the first line where they state:
$$f(x) \equiv x^3 - \sin^3 x = x^3 + {1 \over 4} \,\sin {3x} - {3 \over 4}\,\sin x$$
How ... | HINT: $$\sin{3x}=3\sin {x} -4\sin^3 x \\ \sin {(A+B)}=\sin A \cos B + \cos A \sin B$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/882538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
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.