2
votes
2answers
27 views

Computing $\int_{\gamma} {dz \over (z-3)(z)}$

Compute, using the Cauchy Integral Formula, $$ \int_{\gamma} {dz \over (z-3)(z)} $$ where $\gamma$ is the circle of radius $2$ centered at the origin, oriented counterclockwise. ...
1
vote
4answers
38 views

verifying log rules via elementary properties of an integral

my attempt at this question so far, $$\int_1^x \frac{1}{t}dt +\int_1^y \frac{1}{t}dt= 2\int_1^x \frac{1}{t}dt+ \int_x^y \frac{1}{t}dt$$ But I am not sure hwo to prove it from elementary ...
3
votes
0answers
35 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
7
votes
3answers
678 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
0
votes
0answers
57 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
0
votes
1answer
31 views

Integral-Fourier sum

I am trying to prove the following relation in (3) where $\alpha,\beta,\gamma,\delta,\omega \in \mathbb{R}$. Given the integral $$ I=\frac{1}{2}\int_0^\alpha dx \left( \beta ...
2
votes
1answer
39 views

Double integral and polar coordinates

Please, help me solve this double integral $$\int^{2\pi}_0d\varphi\int^{2}_1\frac{1}{\sqrt{\rho^3\cos^3\varphi+\rho^3\sin^3\varphi}}\rho\,d\rho$$ I really don't know how to figure out and carry of ...
1
vote
1answer
22 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
2
votes
4answers
82 views

Solving the integral $\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$

I really don't know how to solve this integral $$\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$$ Should I use firstly a formula of $\sin(a+b)$?
13
votes
1answer
205 views

Formula for $\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$

Is it possible to express the following integral in terms of known special functions? $$I(a,b)=\int_0^\infty \frac{\log(1+x^2)}{\sqrt{(a^2+x^2)(b^2+x^2)}}dx$$ I have managed to solve the special ...
8
votes
1answer
152 views

Closed Form for $\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$

Is there a closed form for the following integral? $$\int_0^1 \frac{\log(x)}{\sqrt{1-x^2}\sqrt{x^2+2+2\sqrt{2}}}dx$$ It is approximately equal to $-0.48878092308456029189008$. Mathematica is ...
0
votes
1answer
30 views

Din derivatives and fundamental theorem of calculus

I have been looking for some references concerning the fundamental theorem of calculus and Dini derivatives and I did not find it. I would like to know if given a locally Lipschitz function ...
3
votes
0answers
119 views

integral $I=\int_0^\infty x^{\alpha -1}Li_n (-\sigma x) Li_m(-\omega x^r)dx$.

I am trying to calculate an integral that can be expressed in terms of infinite hypergeometric series by using transforms and Residue method, the integral is $$ ...
6
votes
1answer
64 views

Integral $\int_0^\infty \frac{1}{(1+x^m)(1+x^2)}\,dx$ [duplicate]

I saw somewhere that the above integral is equal to $\pi/4$ for all real number $m$. This seems to be surprising. Does anyone have a nice proof?
6
votes
3answers
279 views

Evaluating $\int_{0}^{1}\frac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$

Find this integral $$\operatorname I=\int\limits_{0}^{1}\dfrac{\arcsin{\sqrt{x}}}{x^4-2x^3+2x^2-x+1}\operatorname d\!x$$ My try: let $$f(x)=x^4-2x^3+2x^2-x+1$$ I found ...
0
votes
4answers
119 views

Calculate the value of $\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$

$$\int_0^\frac{\pi}{6} \frac{\cos x \operatorname d\!x}{\sqrt{\frac{1}{4}-\sin^2x}}$$ so $$\lim_{\epsilon->\frac{\pi}{6}} \int^{\epsilon} _{0} \frac{\cos x}{\sqrt{\frac{1}{4} - \sin^2x }} $$ ...
0
votes
1answer
51 views

Validity of an integral inequality

Suppose we have two functions $f(x)$ and $g(x)$ And $f(x)<g(x)$ for all values of $n$ Then for arbitrary $a$ and $b$ (within the range) is it true that $$\int_b^{a}\frac{dx}{f(x)} > ...
0
votes
0answers
33 views

Computing an (iterated) integral

I am having problems to find a closed form depending on $n$ of the following integral. $$ \int_{t_{0}\ <\ w_{1}\ <\ \cdots\ <\ w_{n}\ <\ t}\ {{\rm d}w_{1}\cdots {\rm d}w_{n} \over ...
1
vote
1answer
47 views

Show $\int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0$

let $\rho(x)=\sqrt{x}, \hspace{4mm} \forall x \in \mathbb{R}$ Show : $$ \int_{0}^{\epsilon}\rho(x)^{-2}dx= +\infty, \hspace{4mm} \forall \epsilon >0. $$ My attempt: \begin{align*} ...
8
votes
1answer
114 views

Integral in $n-$dimensional euclidean space

I want to calculate this integral in $n$-dimensional euclidean space. $$I(x)=\int_{\mathbb{R}^n}\frac{d^n k}{(2\pi)^n}\frac{e^{i(k\cdot x)}}{k^2+a^2},$$ where $k^2=(k\cdot k)$, ...
5
votes
1answer
104 views

Integral $\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$ [duplicate]

Consider $$\int_{-1}^{1}\frac{\sqrt{(1-x^{2})}}{1+x^{2}}dx$$ I have a problem with this integral; the method I know consists in calculating the complex integral of $$f(z) = \left( \frac{z-1}{z+1} ...
2
votes
1answer
96 views

Evaluation of a definite integral

I want to find the best way to show $\int_0^\infty\dfrac{x^{2m}}{x^{2n}+1}\,dx=\dfrac{\pi}{2n}\operatorname{csc}\left(\dfrac{2m+1}{2n}\pi\right)$, where $0\leq m<n$. It's easy to verify some ...
1
vote
2answers
157 views

Integral $\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$ via contour integration

I want to evaluate the following integral $$\int_{0}^{1}\frac{1}{x^{2}+2x+2}dx$$ by contour integration; I have a problem with the choice of the contour/ branch cuts. Where can I find some some ...
1
vote
2answers
42 views

evaluation of an integral involving algebraic numbers

Define $A\colon[0,1]\to \mathbb R$ by $A(x) =\begin{cases} 1, &\text{if }x \text{ is algebraic}\\ 0, &\text{otherwise} \end{cases}$ Evaluate : $\int_{0}^{1}A(x)dx$ Is it ...
0
votes
1answer
79 views

Integral calculation of the expected area of intersection of two circles

I want to calculte the following integral: $$ \frac{2 \int _0^1\int _{-\sqrt{1-y_2^2}}^{\sqrt{1-y_2^2}}\int _{-1}^1\left(2 \cos ^{-1}\left(\frac{1}{2} \sqrt{(x_1-x_2)^2+y_2^2}\right) \ -\ \sin ...
22
votes
2answers
441 views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

In the following thread I arrived at the following result $$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$ Defining $$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, ...
2
votes
5answers
148 views

$\int_{0}^{\pi} \exp\left(\cos\left(t\right)\right)\cos\left(\sin\left(t\right)\right)\,{\rm d}t=\pi$

Does anyone have a proof of the above integral? I have one proof, but I wanted to see other proofs.
4
votes
1answer
92 views

Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
4
votes
1answer
105 views

How to integrate the bump functions,i.e,$\int_a^{b}e^{-\frac{1}{x-a}+\frac{1}{x-b}}dx$,where $a<b$.

Since $$\lim_{x\to{a}}e^{-\frac{1}{x-a}+\frac{1}{x-b}}=\lim_{x\to{b}}e^{-\frac{1}{x-a}+\frac{1}{x-b}}=0,$$ $e^{-\frac{1}{x-a}+\frac{1}{x-b}}$ is continuous on the interval $[a,b]$ (taking $0$ if ...
0
votes
1answer
17 views

Which of these compact sets are possible such that they have the following measures?

I am supposed to construct compact sets $K \subset \mathbb{R}$ (if possible) that have the following properties: lambda is the Lebesgue-measure: $ \lambda (K^0) = \lambda (K)$. This is easy, just ...
0
votes
1answer
116 views

Prove $\text{Beta}(x,y) = 2\int_0^{\pi/2}(\sin\theta)^{2x-1}(\cos\theta)^{2y-1}\,d\theta, \qquad \mathrm{Re}(x)>0,\ \mathrm{Re}(y)>0$

Prove that $$\int_0^1 x^{k}(1-x)^kdx=\frac{k!k!}{(2k+1)!}.$$ (Edit: Actually the proof can be found here http://en.wikipedia.org/wiki/Beta_function ) How would you show this $\text{Beta}(x,y) = ...
0
votes
3answers
57 views

Fast way to do this well-known integral (gaussian-distribution)

I want to evaluate $$ \frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...
1
vote
3answers
107 views

How to show the following cool equality:

I am looking for a proof of the following relationship: $\newcommand{\ds}[1]{\displaystyle{#1}}$ $$ \frac{\ds{\int_{0}^{\pi}\sin^{n-2}\left(t\right)\,{\rm d}t}} ...
1
vote
0answers
47 views

How to integrate this function without using the transformation theorem?

I want to integrate the function $\int_\mathbb{R^n} \psi_{\epsilon}(|x|_2) dx$ where we have $\psi_{\epsilon} ( t) = 1$ for $0\le t \le R$ $\psi_{\epsilon} ( t) = 1+\frac{R - t}{\epsilon}$ for $0\le t ...
8
votes
1answer
309 views

Evaluating $\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx = -\frac{\pi^3}{48}-\frac{\pi}{8}\log^2 2 +G\log 2$$ where $G$ is the Catalan's Constant. Numerically, it's ...
9
votes
0answers
233 views

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

How can we prove that: $$\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx=\frac{7\pi^2}{48}\zeta(3)-\frac{25}{16}\zeta(5)$$ where $\zeta(z)$ is the Riemann Zeta Function. The best I could do was ...
1
vote
1answer
44 views

choose relevent length of interval

i have question related to Riemann sum. Is then length of interval matters? Suppose we want to calculate Riemann net sum of function $f(x)=3-x/2$ in this interval $[2,14]$. I first take $n=2$, ...
1
vote
1answer
66 views

Why this Equivalence of integrals is true?

$$\int_b^{\infty}(1+y^{2})e^{-y^2}\left[\frac{f(y)-f(x)}{y-x}\right]^{2}dy=O(1)\int_b^{\infty}e^{-y^2}\left[f^2(y)+f^2(x)\right]dy$$ enter link description here go to pg 72 in the end this pg. I ...
14
votes
2answers
556 views

A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

I encountered a strange integral with a strange result. $$\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx = \pi \log \left(\log (2) \right)$$ Believe it or not, the result agrees ...
49
votes
3answers
3k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( ...
5
votes
3answers
141 views

How to calculate this integral

I have a function $u(x,y)$ defined on the domain $|x|<\infty, y>0$. I know that $$ \frac{\partial u(x,y)}{\partial y} = \frac{y}{\pi}\int_{-\infty}^{\infty} \frac{f(w)}{y^2 + (x-w)^2}dw$$ How ...
3
votes
0answers
33 views

fancy about some properties of kernel functions at infinity

Consider the two common types of kernel functions $\sum\limits_{t=a}^bf(t)K(x,t)$ and $\int_a^bf(t)K(x,t)~dt$ , prove whether the following properties are correct or not: $1.$ If $K(x,t)$ is bounded ...
8
votes
2answers
245 views

What is $\int_0^{\infty}\!e^{-x^2}e^{-ae^{bx^2}}\,dx$?

I've been trying without success to evaluate $$ {\Large\int_0^{\infty}\!e^{-x^2}e^{\Large \,-ae^{\,bx^2}}\,dx}. $$ It's not in my integral tables. Wolfram online integrator won't do it. It doesn't ...
7
votes
2answers
416 views

Integral $\int_0^\pi \cot(x/2)\sin(nx)\,dx$

It seems that $$\int_0^\pi \cot(x/2)\sin(nx)\,dx=\pi$$ for all positive integers $n$. But I have trouble proving it. Anyone?
3
votes
1answer
355 views

Multiple Choice question about a continuous function

If $f \colon \mathbb R\to\mathbb R$ is a continuous function, which of the following statements implies that $f(0)=0$? (A) $\int_0^1 f(x)^n \,dx\to 0$ as $n\to\infty$ (B) $\int_0^1 f\left(\frac ...
1
vote
1answer
138 views

Evaluate the integral $\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$.

Evaluate: $$\int_{-1}^1\frac{dx}{(e^x+1)(x^2+1)}$$
1
vote
1answer
399 views

Limit of $ (\int_a^b f(x)^n \ dx)^{1/n}$ when $n\to\infty$

I've been working on the following problem: Show that if $f\in C[a, b]$ , $f\ge 0$ on $[a, b]$, then $\left(\int_a^b f(x)^n \,dx\right)^{1/n}$ converges when $n\to\infty$ and the limit is ...
1
vote
2answers
191 views

Numerical integration given a derivative of a function of two dependent variables

I want to solve the following equation of an integral valued function: $Q = \int_{0}^{x_p}f(t_p,x)dx$ for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. ...
5
votes
1answer
139 views

What is the general solution for integrals of the form $\int_{0}^{\infty}\frac{x^{2 m}\;\ln^{n}(x) }{e^{\frac{2 p +1}{2}x}} dx$?

I have this integral $$\int_0^\infty \frac{x^{2 m}\;\ln^n (x) }{e^{\frac{2 p +1}{2}x}} dx\;\; m,n,p \in \mathbb{N}$$ and I'like to know the general solution. From WolframAlpha I got some ...
1
vote
1answer
157 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...