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Zaid Alyafeai's user avatar
Zaid Alyafeai's user avatar
Zaid Alyafeai's user avatar
Zaid Alyafeai
  • Member for 10 years, 9 months
  • Last seen more than 3 years ago
33 votes

Really advanced techniques of integration (definite or indefinite)

21 votes

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

20 votes
Accepted

Double Integral $\int_0^\infty \int_0^\infty \frac{\log x \log y}{\sqrt {xy}}\cos(x+y)\,dx\,dy=(\gamma+2\log 2)\pi^2$

20 votes
Accepted

How can we show that $\int_{0}^{1}\cos(\ln x)\cdot{\mathrm dx\over 1+x^2}={\pi\over 4}\cdot{1\over \cosh\left({\pi\over 2}\right)}?$

19 votes

What is ML Inequality property of complex integral

13 votes

Evaluate the integral $\int_{0}^{+\infty}\frac{\arctan \pi x-\arctan x}{x}dx$

13 votes
Accepted

On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$

13 votes
Accepted

A closed form for a triple integral with sines and cosines

13 votes
Accepted

Evaluating $\int_0^1 \frac{\ln^m (1+x)\ln^n x}{x}\; dx$ for $m,n\in\mathbb{N}$

13 votes

How can we show that $\int_{-\infty}^{+\infty}{ke^x\pm1\over \pi^2+(e^x-x+1)^2}\cdot{(e^x+1)^2\over \pi^2+(e^x+x+1)^2}\cdot 2x \,\mathrm dx=k?$

11 votes
Accepted

How may one show that $\int_{0}^{\pi/2}{\ln\cos x\over \tan x}\cdot\ln\left({\ln\sin x\over \ln \cos x}\right)\mathrm dx={\pi^2\over 4!}?$

11 votes

Some users are mind bogglingly skilled at integration. How did they get there?

11 votes

Great books on all different types of integration techniques

11 votes

Integral $\int_0^\infty \log(1+x^2)\frac{\cosh{\frac{\pi x}{2}}}{\sinh^2{\frac{\pi x}{2}}}\mathrm dx=2-\frac{4}{\pi}$

11 votes
Accepted

What is the residue of $f(z)=\tan{z}$ at any of its pole ? Is the solution correct?

10 votes

Evaluate $ \int_{0}^{1} \ln(x)\ln(1-x)\,dx $

10 votes

How to integrate: $\int_0^{\infty} \frac{1}{x^3+x^2+x+1}dx$

10 votes

Proof only by transformation that : $ \int_0^\infty \cos(x^2) dx = \int_0^\infty \sin(x^2) dx $

9 votes
Accepted

An integral of rational function with third power of cosine hyperbolic function

8 votes
Accepted

Closed form of $\int_{0}^{\infty}\sum\limits_{k=0}^{n}{{n\choose k}e^{-(2k+1)x}\over \cosh^n(x)}\mathrm dx$

8 votes
Accepted

Help find closed form for:$\sum_{n=0}^{\infty}{(-1)^n\left({{\pi\over 2}}\right)^{2n}\over (2n+k)!}=F(k)$

8 votes
Accepted

What is the closed form of $\sum_{n\geq 1}(-1)^{n-1}\psi'(n)^2$?

8 votes
Accepted

Proof involving gamma function, infinite product and Gauss

8 votes
Accepted

Integral of $\frac{x^\alpha}{(1 + x^2)^2}$ from 0 to $\infty$ using contour integration for $-1 < \alpha <3$

7 votes

How can we show that $\int_{0}^{1}\cos(\ln x)\cdot{\mathrm dx\over 1+x^2}={\pi\over 4}\cdot{1\over \cosh\left({\pi\over 2}\right)}?$

7 votes

On twisted Euler sums

7 votes

Does $\sum _{k=2} ^\infty \frac{(-1)^k}{\sqrt{k}+(-1)^k}$ converge conditionally?

7 votes

How to evaluate $\sum_{n=1}^{\infty}\frac{H^3_{n}}{n+1}(-1)^{n+1}$.

7 votes
Accepted

prove that: $\sqrt{2}=e^{1-{2K\over \pi}}\prod\limits_{n=1}^{\infty}\left({4n-1\over 4n+1}\right)^{4n}e^2$

7 votes
Accepted

How to evaluate $I=\int_0^{\frac{\pi}{2}}\sin^2x\ln(\sin^2(\tan x))dx$

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