Given....int cos^2x
from Euler's substitution........ cosx = [e^{ix} + e^{-ix} ]/2
int cos^2x = int [ (e{ix} + e^{-ix})/2 ]^2dx
= 1/4int [e^{2ix} + 2 + e^{-2ix}]dx
=1/4 int e^{2ix} dx + int 1/2 dx +1/4 int e^{-2ix} dx
=1/4[ (e^{2ix} - e^{-2ix})/2i ] + (1/2)x + c
But
(e^{2ix} + e^{-2ix})/2i = sin(2x)
therefore
int cos^2(x) = 1/4{sin(2x)} + (1/2)x + c