the answer is correct...
but is there no other way...
it is like round up.
use integration by parts and let u=cosx and dv=cosxdx
du=-sinx v=sinx
Int(udv)=uv-Int(vdu) so uv=cosx(sinx) and vdu=sinx(-sinx)
so we have:
Int(cos^2(x)=(cosx)(sinx)+Int(sin^2x)
(the (-) became + because of the -sinx, so we add Int(vdu))
Now it looks not better because we have sin^2x instead of cos^2x,
but sin^2x=1-cos^2x since sin^2x+cos^2x=1
So we have
Int(cos^2x)=cosxsinx+Int(1-cos^2x)
=cosxsinx+Int(1)-Int(cos^2x)
So now add the -Int(cos^2x) on the RHS to the one on the LHS
2Int(cos^2x)=cosxsinx+x
so Int(cos^2x)=1/2[cosxsinsx+x] and now add the constant!
final answer
Integral of cos^2x=(1/2)(cosx sinx + x)+C = x/2 + (1/4)sin 2x + C
(because sin x cos x = (1/2)sin 2x)
right??