7
votes
1answer
113 views

Evaluate $\int_{0}^{\large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\frac{\ln{(\sin x)}}{\cot x}+\frac{\ln {(\cos x)}}{\tan x}\right)dx$

How do I find the value of this integral? $$I=\int_{0}^{\Large\frac{\pi}{4}} \ln {(\sin x)}\cdot\ln {(\cos x)} \left(\dfrac{\ln{(\sin x)}}{\cot x}+\dfrac{\ln {(\cos x)}}{\tan x}\right)dx$$ I tried ...
1
vote
0answers
27 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
3
votes
2answers
128 views

Evaluating $\int^b_a \frac{dx}{x}$ from the definition of the integral

I know that $$\int^b_a \frac{dx}{x}=\ln b-\ln a$$ I'm trying to evaluate this integral using the same method used in this answer: http://math.stackexchange.com/a/873507/42912 My attempt $\int^b_a ...
6
votes
2answers
183 views

Integral $\int_{0}^{\pi/2} \arctan \left(2\tan^2 x\right) \mathrm{d}x$

The following integral may seem easy to evaluate ... $$ \int_{0}^{\Large\frac{\pi}{2}} \arctan \left(2 \tan^2 x\right) \mathrm{d}x = \pi \arctan \left( \frac{1}{2} \right). $$ Could you prove ...
0
votes
2answers
34 views

Convergence of improper integral of $\ln f(x)$

Is there something know about the convergence of $\int_0^1 \ln f(x)dx $ for $f(x)$ continous on $\left(0,1\right)$ and both limits exists, i.e. $\lim_{x\to 0} f(x)$ and $\lim_{x\to 1} f(x)$ ? I ...
8
votes
2answers
190 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
7
votes
2answers
197 views

Proving that $\int_0^1\frac{x \log^2(1-x)}{1+x^2} \ dx = \frac{35}{32}\zeta(3)+\frac{1}{24}\log^3(2) -\frac{5}{96} \pi^2 \log(2)$

Could we possibly prove this result without using the polylogarithm? I know how to do it by polylogarithm means, but I want a different way. Is that possible? $$\int_0^1\frac{x \log^2(1-x)}{1+x^2} ...
6
votes
3answers
233 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
9
votes
0answers
153 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
11
votes
3answers
342 views

Finding the maximum value of $\displaystyle \int_{0}^{1}e^x\log f(x)dx$ when $\displaystyle \int_{0}^{1}f(x)dx=1$

Suppose that $f(x)\ (0\le x\le 1)$ is continuous and strictly positive and satisfies $$\int_{0}^{1}f(x)dx=1.$$ Then, can we find the maximum and the minimum value of the following? If yes, then how? ...
5
votes
2answers
156 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
1
vote
1answer
37 views

defenite integral involve bessel function

I have an integral which involves Bessel function as follows: $I=\int_{r=0}^a \int_{\theta=0}^{2\pi}(e^{-jkr\cos(\theta-\phi)}d\theta)rdr$ I have tried with $e^{-jkr\cos(\theta-\phi)}=\sum ...
1
vote
2answers
98 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
8
votes
1answer
256 views

Evaluation of $\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$

I need some hints, clues for getting the closed form of $$\int_0^1 \frac{\log(1+x)}{1+x}\log\left(\log\left(\frac{1}{x}\right)\right) \ dx$$
0
votes
1answer
30 views

Finding volume between plane and paraboloid

Find the volume between bounded by $z=4$ and $z=x^2+y^2$.(Answer: $8\pi$) I thouhg using dievergence theorm ($\iint_KdivFdxdydz=\iint_SF\cdot\hat{n}dS$) for $\vec{F}=\big(\frac x 2,\frac y ...
4
votes
3answers
264 views

Integral without residues

How do I do this integral without using complex variable theorems? (i.e. residues) $$\lim_{n\to \infty} \int_0^{\infty} \frac{\cos(nx)}{1+x^2} \, dx$$
4
votes
3answers
524 views

Prove the equation

Prove that $$\int_0^{\infty}\exp\left(-\left(x^2+\dfrac{a^2}{x^2}\right)\right)\text{d}x=\frac{e^{-2a}\sqrt{\pi}}{2}$$ Assume that the equation is true for $a=0.$
3
votes
2answers
102 views

Show that $\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$

I wish to show $$\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$$ I've tried substitution and integration by parts to get a recursive formula for ...
18
votes
2answers
278 views

A closed form for a lot of integrals on the logarithm

One problem that has been bugging me all this summer is as follows: a) Calculate $$I_3=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1} \ln{(1-x)} \ln{(1-xy)} \ln{(1-xyz)} \,\mathrm{d}x\, \mathrm{d}y\, ...
0
votes
2answers
53 views

Finding $(p,q)$ such that $\frac{x^p}{1+x^q}$ is integrable on $(0,+\infty)$

I'm trying to show that $f(x) = \frac{x^p}{1+x^q}$ is integrable on $(0,\infty)$ if and only if $p > -1$ and $q-p > 1$. So on $[1,\infty)$ we can compare with $g(x) = x^{p-q}$ which is ...
6
votes
4answers
152 views

How can we prove $\int_1^\pi x \cos(\frac1{x}) dx<4$ by hand?

Is there any way we can prove this definite integral inequality by hand: $$ \int_{1}^{\pi}x\cos\left(1 \over x\right)\,{\rm d}x < 4 $$ I don't where to start even, please help. That ...
2
votes
1answer
51 views

If the integral of a non-negative function is $0$, then the function is $0$

Suppose that $f$ is a continuous function on $[a,b]$ and that $f(x)\geq0$ for all $x\in [a,b]$. Show that if $\int_a^bf(x)=0$, then $f(x)=0$ for all $x\in[a,b]$. Let $F(x)=\int_a^xf(x)$. Since ...
3
votes
0answers
95 views

The elementary methods to compute $\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\quad;\quad\text{for}\, \alpha>0$

How to compute the following integral using elementary methods (high school methods). \begin{equation}\int_0^\pi\frac{e^{ix}}{x-\alpha e^{ix}}\,dx\qquad;\qquad\text{for}\, ...
1
vote
3answers
102 views

Evaluate $\int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\,dx\,dy$

I would like to compute the following, $$ \int_0^{\infty}\int_0^{\infty}e^{-x^2-2xy-y^2}\ dx\,dy $$ It is obvious that we can rewrite the integral above to, $$ ...
3
votes
3answers
172 views

Integral $\int_0^{\infty} \frac{x^{a-1}}{1+x} dx $ converges?

For what values โ€‹โ€‹of $a \in \mathbb{R}$ the following integral converges? $$\int_0^{\infty} \frac{x^{a-1}}{1+x}\ dx $$ I tried to compute the integral but I stuck solving and then I tried to compare ...
0
votes
2answers
44 views

Integrability of a piecewise function [closed]

I got this question: Is the function $g$ defined below is integrable on $[0,1]$? $$g(x)= \begin{cases} \dfrac{1}{x}\sin\dfrac{1}{x} & \text{if $x\neq 0$} \\[8pt] 3 & \text{if $x=0$} ...
-1
votes
1answer
29 views

condition for convergence to Riemann integral

Suppose that functions f and g are defined on [0, T] and bounded. Let [0,T] is divided into n point. that is , $t_1=0<t_1=T/n<...<T_n=T$ $\Sigma f(t_i)g(t_i)$ i want to show that as n ...
13
votes
2answers
234 views

The 3 Integral $\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x-1})^2}}=\frac{\pi}{3\sqrt 3}\big(\log 3-\frac{\pi}{3\sqrt 3} \big)$

Hi I am trying evaluate this integral and obtain the closed form:$$ I:=\int_0^\infty \frac{x\,dx}{\sqrt[\large 3]{(e^{3x}-1)^2}}=\frac{\pi}{3\sqrt 3}\left(\log 3-\frac{\pi}{3\sqrt 3} \right). $$ The ...
13
votes
2answers
423 views

Integral $\int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma x)\,dx$

I am having trouble showing this equality is true$$ \int_0^\infty \frac{\sqrt{\sqrt{\alpha^2+x^2}-\alpha}\,\exp\big({-\beta\sqrt{\alpha^2+x^2}\big)}}{\sqrt{\alpha^2+x^2}}\sin (\gamma ...
5
votes
2answers
120 views

Integral $\int_0^{\pi/4}\frac{x^2\tan x}{\cos^2 x}dx=\frac{\log 2}{2}-\frac{\pi}{4}+\frac{\pi^2}{16}$

Hi I am trying to evaluate the definite integral which has a closed form given by: $$ \mathcal{I}=\int_0^{\pi/4}\frac{x^2\tan x}{\cos^2 x}dx=\frac{\log 2}{2}-\frac{\pi}{4}+\frac{\pi^2}{16}. $$ We can ...
1
vote
1answer
39 views

$\lim_{n\to \infty}\sum^n_{i=1}\frac{2}{\sqrt{4n^2-i^2}}$

How to calculate the following limit? $\lim_{n\to \infty}\sum^n_{i=1}\frac{2}{\sqrt{4n^2-i^2}}$ I think it should be a definite integral, but what's the function to integrate and over what interval? ...
12
votes
3answers
220 views

Compute $\int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx$

How to compute the following integral \begin{equation} \int_0^\pi\frac{\cos nx}{a^2-2ab\cos x+b^2}\, dx \end{equation} I have been given two integral questions by my teacher. I cannot answer ...
2
votes
2answers
66 views

Integral involving a logarithm and a linear rational function

$$\int_{0}^{1} \frac{\log x}{x-1}dx$$ I was wondering: is it possible to evaluate this integral with real methods? Playing around with a series expansion I was able to recognize that the integral is ...
3
votes
2answers
66 views

I want to prove $k(x,t)=\frac{1}{\sqrt{4\pi t} } e^{\frac{-x^2}{4t}} $

I have this integral $$u(x,t)=\int _{-\infty}^{\infty} f(\eta)\left[\frac{1}{2\pi}\int _{-\infty}^{\infty}e^{iw(x-\eta)-w^2t}\ dw\right]\ d\eta=\int _{-\infty}^{\infty}k(x-\eta,t)f(\eta)\ d\eta$$ I ...
2
votes
3answers
130 views

How to integrate $\int_0^\pi \frac{1}{\sqrt{1+k^2\sin^2\phi}} d \phi$?

I am currently dealing with the integral $$\int_{0}^{\large\pi}\frac{{\rm d}\phi} {\,\sqrt{\vphantom{\Large A}\,1 + k^{2}\sin^{2} \phi \,}\,} $$ I know that if I had a minus sign in the denominator, ...
6
votes
3answers
156 views

Integral $\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\frac{17\pi^4}{360}$

Hi I am trying to integrate $$ \mathcal{I}:=\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\int_0^1\int_0^1 \frac{\log(1-x)\log(1-y)}{1-xy}dx \,dy $$ A closed form does ...
0
votes
1answer
58 views

Definite integral with difficult term in the denominator.

I want to integrate $$\int_0^{2\pi} \frac{1}{\sqrt{r^2+R^2-2rR \cos(\phi)+(z-b)^2}}-\frac{1}{\sqrt{r^2+R^2-2rR \cos(\phi)+(z+b)^2}} d \phi.$$ You may assume that the denominator is not zero, as $z ...
5
votes
4answers
201 views

Integral $\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi$

Hi I am trying to prove this result below $$ \mathcal{J}:=\int_0^1\frac{dx}{\sqrt{\log \frac{1}{x}}}=\sqrt \pi. $$ The result is quite interesting however I realized I am not familiar with working ...
2
votes
2answers
59 views

What is the difference to compute double integral?

I see a double integral on web with a strange way to calculate. Please help me to make it clear. Here is the integral: $$\int_1^2\int_1^2(x+y)\ dx\ dy$$ As my way, I calculate it: ...
10
votes
3answers
156 views

Integral $\int_0^\infty \log(1-e^{-a x})\cos (bx)\, dx=\frac{a}{2b^2}-\frac{\pi}{2b}\coth \frac{\pi b}{a}$

$$\mathcal{J}:=\int_0^\infty \log(1-e^{-a x})\cos (bx)\, dx=\frac{a}{2b^2}-\frac{\pi}{2b}\coth \frac{\pi b}{a},\qquad \mathcal{Re}(a)>0, b>0. $$ I tried to write $$ \mathcal{J}=-\int_0^\infty ...
10
votes
1answer
142 views

Integral $\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3$

Hi I am trying to prove this below. $$ I:=\int_0^\infty \frac{\cos x}{x}\left(\int_0^x \frac{\sin t}{t}dt\right)^2dx=-\frac{7}{6}\zeta_3 $$ where $$ \zeta_3=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I am ...
6
votes
3answers
161 views

Integral $\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a}$

I am trying to prove this interesting integral$$ \mathcal{I}:=\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a},\qquad \mathcal{Re}(a)\neq 0. $$ This result is breath taking but I ...
3
votes
3answers
177 views

Integral with binomial coefficient

Is it possible to evaluate this integral without using the gamma function $$ \int_0^1 {a \choose b}x^b(1-x)^{a-b} dx$$ It looks a little like part of binomial theorem, but I don't have an idea how to ...
2
votes
2answers
128 views

Integral $\int_0^\infty \frac{\sin a x-\sin b x}{\cosh \beta x}\frac{dx}{x}$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\sin a x-\sin b x}{\cosh \beta ...
4
votes
1answer
151 views

$\int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2$

Hi I am trying to prove this $$ \int_0^\infty \frac{x\cos ax}{1+x^2}\coth \frac{\pi x}{4} dx=\frac{\pi}{2}e^{-a}+\cosh a\log \coth \frac{a}{2}+2\sinh a \arctan(e^{-a})-2,\qquad a>0. $$ What a ...
3
votes
2answers
112 views

$\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\big( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh \frac{a\pi}{2\beta}}\big)$

Hi I am trying to prove this interesting integral $$ \mathcal{I}:=\int_0^\infty \frac{\cos a x-\cos b x}{\sinh \beta x}\frac{dx}{x}=\log\left( \frac{\cosh \frac{b\pi}{2 \beta}}{\cosh ...
2
votes
3answers
158 views

Integral $\int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}$

EDIT: Small Typo Fixed now, Thanks to Sir Chen Wang! Hi I am trying to prove this result without using a series approach $$ \int_0^1\log(1+x)\frac{1+x^2}{(1+x)^4}dx=-\frac{\log 2}{3}+\frac{23}{72}. ...
13
votes
3answers
605 views

Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$ [duplicate]

Prove the following \begin{equation}\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx=\frac{\pi}{4}+\frac{\pi}{4e^2}\end{equation} I would love to see how Mathematics SE users prove the integral ...
3
votes
0answers
54 views

Integral vs antiderivative

I have a similar question to this one: Integrable or antiderivative. If a function has an antiderivative, does the difference of values of the antiderivative on the endpoints of an interval always ...
10
votes
6answers
259 views

Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}dx=\frac{10\pi^3}{81 \sqrt 3}$

Hi how can we prove this integral below? $$ I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}dx=\frac{10\pi^3}{81 \sqrt 3} $$ I tried to use $$ I=\int_0^1 \frac{\log^2x}{1-x(1-x)}dx $$ and now tried changing ...