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A collection of definite integrals

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Several well known definite integrals can be found in the book Table of integrals series and products by I.S. Gradshteyn and I.M. Ryzhik. In this article, I present a few definite integrals which I could not find the this book. This work on definite integral is inspired by Ramanujan’s legendary work in definite integrals. I must say that finding beautiful integrals is definitely addictive.

Entry 1. If G denotes the Catalan constant and \gamma denotes the Euler–Mascheroni constant then

\displaystyle{\int_{0}^{\infty}e^{-x^4}\ln^2 x dx = \frac{\Gamma(\frac{1}{4})}{256}\{32G+4\gamma^2+5\pi^2+36\ln^2 2}

\displaystyle{+12\pi\ln2+4\gamma\pi+24\gamma\ln2\}}.

Entry 2. If Im(a) >0, b>0 then

\displaystyle{\int_{0}^{\infty} i^{2ax^b}dx = \frac{1}{b}\Gamma\Big(\frac{1}{b}\Big)\Big(\frac{i}{a \pi}\Big)^{1/b}}.

Entry 3. If Re(a) >0, b>1, c>0, d>0 then

\displaystyle{\int_{0}^{\infty} \frac{e^{-ax^b}}{1+dx^b} dx = \frac{e^{a/d}}{d^{1/b}}\Gamma\Big(\frac{b+1}{b}\Big)\Gamma\Big(\frac{b-1}{b}, \frac{a}{d}\Big)}.

Entry 4.

\displaystyle{\int_{0}^{\infty} \frac{e^{-ax^{-b}}}{1+dx^b} dx = \frac{e^{ad}}{d^{1/b}}\Gamma\Big(\frac{b-1}{b}\Big)\Gamma\Big(\frac{1}{b}, \frac{a}{d}\Big)}.

Entry 5.

\displaystyle{\int_{0}^{\infty} \frac{e^{-ax^b}}{1+dx^{-b}} dx = \frac{e^{ad} d^{1/b}}{b^2}\Gamma\Big(\frac{1}{b}\Big)\Gamma\Big(-\frac{1}{b}, ad\Big)}.

Entry 6. If a > 0, b > 0 then

\displaystyle{\int_{0}^{\infty} \ln\Big(1 + \frac{a^2}{x^2}\Big) \cos (bx) dx = \frac{\pi - \pi e^{-ab}}{b}}.

Entry 7. If G denotes the Catalan constant then

\displaystyle{\int_{0}^{1} \Big(\frac{\tan ^{-1} x}{x}\Big)^2dx = G - \frac{\pi^2}{16} + \frac{\pi \ln 2}{4}}.

Entry 8.

\displaystyle{\int_{0}^{1} (\tan ^{-1} x)^2dx = -G + \frac{\pi^2}{16} + \frac{\pi \ln 2}{4}}.

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