# New post in Closed-form

## Improper Integral $\int_0^1\left(\left\{\frac1x\right\}-\frac12\right)\frac{\log(x)}xdx$

September 9th, 2020

June 6th, 2021

August 1st, 2022

May 25th, 2020

## How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$

October 26th, 2021

August 1st, 2022

August 1st, 2022

August 1st, 2022

August 1st, 2022

August 1st, 2022

## Prove $\int_0^{\infty }\frac1{\sqrt{x}}\left(\frac{\cos(\pi x^2)}{\sinh (\pi x)}-\frac1{\pi x}\right)dx=\frac{1}{\sqrt{2}}\zeta(\frac{1}{2})$

December 6th, 2020

March 18th, 2020

August 1st, 2022

## Closed form for improper definite integral involving trig functions and exponentials?

August 23rd, 2020

July 12th, 2020

August 1st, 2022

August 1st, 2022

August 1st, 2022

## What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}}$?

November 24th, 2020

May 5th, 2020

August 1st, 2022

August 1st, 2022

August 1st, 2022

August 1st, 2022

## Prove $\int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2$

August 13th, 2020

## Integral of Combination Log and Inverse Trig Function

September 18th, 2020

## Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$

January 11th, 2022

## Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

October 12th, 2020

## A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

October 10th, 2020

August 1st, 2022