cos^2(2x)?
I = integral
Icos^2(2x)dx= (1/2)Icos^2(t)dt
2x = t, 2dx = dt, dx = dt/2
Icos^2(t)dt= I costcostdt
u=cost, du = - sint
dv = costdt, v = sint
Icos^2(t)dt = sintcost +Isin^2(t)dt
Icos^2(t)dt =sin(2t)/2+I(1-cos^2(t))dt
Icos^2(t)dt =sin(2t)/2 + Idt - I cos^2(t)dt
Icos^2(t)dt +Icos^2(t)dt =sin(2t)/2+
2Icos^2(t)dt =sin(2t)/2 + t
Icos^2(t)dt = sin(2t)/4 + t/2
t=2x
= sin(4x)/4 + x