Linked Questions

6
votes
1answer
784 views

Evaluate $\int_0^\infty \frac{\ln^2(z)}{1+z^2}$dz by contour integration [duplicate]

Background: This is part b of problem 12.4.3 from Arfken, Weber, Harris Math Methods for Physicists to show that $\int_0^\infty \frac{\ln^2(z)}{1+z^2}$dz$=4(1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+...
3
votes
0answers
188 views

Evaluating the definite integral $\int_0^{\infty} \frac{(\ln x)^2}{x^2 + 1} dx$ [duplicate]

I have two integral questions listed below: $$\int_0^{\infty} \frac{\ln x}{x^2 + 1} dx \qquad (1)$$ $$\int_0^{\infty} \frac{(\ln x)^2}{x^2 + 1} dx \qquad (2)$$ The first one, I've solved it, ...
0
votes
0answers
70 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + e^u}...
1
vote
0answers
57 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}=\frac{d^2}{d\...
86
votes
10answers
6k views

Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: $$\...
19
votes
5answers
3k views

Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$

After numerical analysis it seems that $$ \frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}} $$ Could someone prove the validity of such identity?
5
votes
2answers
588 views

Using complex analysis to evaluate $\int_0^\infty\frac{(\ln x)^3}{1+x^2}d x$

Here is my attempt: Let $R>1>r$ and $C$ be the closed curve in $\mathbb{C}$ consists of the following pieces: $$C_1=\{Re^{it}: t\in(0,\pi)\},\quad C_2=[r,R],\quad C_3=\{re^{it}: t\in(0,\pi)\},\...
4
votes
2answers
170 views

Integral along a contour is $0$, how?

I recently had an extremely failed attempt at asking the same question, so I am posting the same question more or less to hope that someone can give me feedback. Consider the integral: $$\int_{0}^{\...
5
votes
1answer
357 views

Using a contour integral to evaluate $\int^\infty_0 \frac{(\log(x))^2}{x^2+1}\,dx $

Say we want to evaluate $\int^\infty_0 \frac{(\log(x))^2}{x^2+1}\,dx $ via integrating over a complex contour. My lecturer said to use the function $f(z) = \frac{(\log(z) - \frac{i \pi}{2})^2}{z^2+1}\...
1
vote
1answer
255 views

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, $\...
1
vote
0answers
197 views

Planning to integrate $\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$ using complex analysis [duplicate]

This is just a plan-out. I want to evaluate: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using a keyhole contour a semi-circle, with base at the x-axis. First I must pick a branch. ...
2
votes
0answers
129 views

Evaluation of $\int_0^\infty\frac{-\ln{x}}{x^2+1}\ln\left(1-\frac{\ln{x}}{x^2+1}\right)\,dx$?

The numerical approximation for this intergral is $$\eqalign{J&=-\int_0^\infty\frac{\ln{x}}{x^2+1}\ln\left(1-\frac{\ln{x}}{x^2+1}\right)\,dx\\&\approx0....