Considering that:
(dsinx)/dx = cosxdsinxdx=cosx
(dcosx)/dx = -sinxdcosxdx=−sinx
int cos^2xdx =int cosx * cosx dx =int cosx d(sinx)∫cos2xdx=∫cosx⋅cosxdx=∫cosxd(sinx)
Integrating by parts:
int cos^2xdx = sinxcosx - int sinx dcosx = ∫cos2xdx=sinxcosx−∫sinxdcosx=
= sinxcosx + int sin^2x dx = =sinxcosx+∫sin2xdx=
= sinxcosx + int (1-cos^2x) dx = =sinxcosx+∫(1−cos2x)dx=
= sinxcosx + x - int cos^2x dx =sinxcosx+x−∫cos2xdx
So:
2int cos^2xdx = sinxcosx + x2∫cos2xdx=sinxcosx+x
and finally:
int cos^2xdx = 1/2(x+1/2sin2x) + C∫cos2xdx=12(x+12sin2x)+C
An alternative method is to use the identity:
cos(2x) = cos^2x-sin^2x = cos^2x - (1-cos^2x) = cos(2x)=cos2x−sin2x=cos2x−(1−cos2x)=
= 2cos^2x-1=2cos2x−1
so that:
cos^2x = (1+cos2x)/2cos2x=1+cos2x2
int cos^2xdx = int (1+cos2x)/2 dx = 1/2(x+1/2sin2x) + C∫cos2xdx=∫1+cos2x2dx=12(x+12sin2x)+C
