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Definite integrals solvable using the Feynman Trick

Nov 15, 2018

How to evaluate $\int\frac{1+x^4}{(1-x^4)^{3/2}}dx$?

Jul 21, 2018

Show that $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$

Jan 5

Prove $\int_0^1\frac{1}{\sqrt{1-x^2}}\arccos\left(\frac{3x^3-3x+4x^2\sqrt{2-x^2}}{5x^2-1}\right)\mathrm dx=\frac{3\pi^2}{8}-2\pi\arctan\frac12$.

Apr 1

Find $\int_1^a \sqrt[5]{x^5-1}\ dx + \int_0^b \sqrt[5]{x^5+1}\ dx$, where $a^5-b^5 = 1$

Dec 29, 2019

Questions

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Proving that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Dec 28, 2019

Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$

Dec 23, 2018

Integral for the New Year $2019$!

Dec 31, 2018

Relation between the roots of a function

Feb 21, 2018

Integral $\int_0^\infty \frac{\ln x}{(\pi^2+\ln^2 x)(1+x)^2} \frac{dx}{\sqrt x}$

Jan 1, 2019

Reputation

+10

Integral $\int_{0}^\frac{\pi}{2} \ln(1+a\sin x)\csc(x)dx $

+10

Proof of this integral identity $\int_0^a{\frac{\tan^{-1}{\left(\sqrt{\frac{2a^2-x^2}{4a^2-x^2}}\right)}}{\sqrt{4a^2-x^2}}dx} = \frac{\pi^2}{32}$

+10

Prove ${_4 F_3} \left(\frac12, \frac12, 1, 1; \frac34, \frac54, \frac32; \frac14 \right)= \frac14 \left(\frac{\pi^2}{4}+\log^2 (2+\sqrt{3} ) \right)$

+10

Integral $\int_0^1 \frac{\ln(1+x+x^2)\ln(1-x+x^2)}{x}dx$

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