I don't know where to start with the following. None of the methods I try seem to get me very far.
Show that $\displaystyle \int_0^{\frac{\pi}{2}} \frac{1}{3+5\cos{x}}dx = \frac{1}{4}\ln{3}$
Use Weierstrass substitution :
$\displaystyle t=\tan \frac x2$
Then $\displaystyle \cos x=\frac{1-t^2}{1+t^2}$
And we have $\displaystyle dx=\frac{2}{1+t^2} ~dt$