Here's two integrals I ran into I thought y'all might like a go. Perhaps they are cliche?.

Show that:

This one has its own name. The Ahmed Integral.

$\displaystyle \int_{0}^{1}\frac{tan^{-1}}{x(x^{2}+1)}dx=\frac{\text{Catalan}}{2}+\frac{\ pi}{8}ln(2)$

In case, $\displaystyle \text{Catalan}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{(2k+1)^{2}}\approx .915....$

The other:

$\displaystyle \int_{0}^{\infty}\frac{sin(x)}{x^{p}}dx=\frac{\sqr t{\pi}{\Gamma}(1-\frac{p}{2})}{2^{p}{\Gamma}(\frac{1}{2}+\frac{p}{2 })}$

or some equivalent form.

I thought these were cool.