Here are some challenging integrals to evaluate.

$(1)$ Evaluate $\displaystyle \int \sin(101x)\sin^{99}x \, dx$. (MIT Integration Bee)

$(2)$ Evaluate $\displaystyle \int^{1}_{0}\frac {\arctan x}{x + 1} \, dx$.

$(3)$ For $n > 1$, prove that $\displaystyle \int^{\infty}_{0}\frac {dx}{\left(x + \sqrt {1 + x^{2}}\right)^{n}} = \frac {n}{n^{2} - 1}$.

$(4)$ If $f$ is a bounded non-negative function, then show that $\displaystyle \int_{0}^{\infty}f(x + \frac {1}{x})\frac {\ln(x)}{x}dx = 0$.

$(5)$ Evaluate $\displaystyle \int_{0}^{1}\ln(\sqrt {1 - x} + \sqrt {1 + x})dx$.

$(6)$ Evaluate$\displaystyle \int_{ - \pi /2}^{\pi /2} \frac1{2007^{x} + 1}\cdot \frac {\sin^{2008}x}{\sin^{2008}x + \cos^{2008}x} \, dx$.

The problems above aren’t necessarily in increasing order of difficulty; however, the last one can be almost impossible to evaluate if one doesn’t know the right “trick”, which will be the subject of my third identity in my series of posts titled, A few useful identities related to definite integrals, which you can find in the Problem Corner.

Note: Most of the problems (posted above) have been borrowed from people who posted the same in AoPS.