There a lot of integrals, however many people solved it in different ways, we can find interesting integrals in Table of Integrals, Series, and Products. I wonder What is the most exciting integral that you solved, created or found it, if you can, also write the solution of the integral, we can find very nicely methods.
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4I recently found a nice little book by V.H. Moll, Special Integrals of Gradshteyn and Ryzhik: the Proofs. I think you'd like it very much, it basically covers exactly what you're asking! – krvolok Apr 22 '15 at 06:45
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2@krvolok Nice!, This book is also awesome: Inside Interesting Integrals, but if you have a favorite integral, please write it. – Apr 22 '15 at 06:55
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@krvolok Nice book! :) – Hiroto Takahashi Jun 09 '16 at 01:03
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$$\int_{0}^{1} \frac{\ln(x^2+1)}{x^2+1}$$ I really like this. It seems to be posted again and again. Solutions can be found here: Evaluate the integral: $\int_{0}^{1} \frac{\ln(x+1)}{x^2+1} \mathrm dx$