Given: #int cot (x) cos^2 (x) dx#
Substitute #cos^2(x) = 1 - sin^2(x)#:
#int cot (x) cos^2 (x) dx= int cot(x)(1-sin^2(x))dx#
Distribute the #cot(x)#:
#int cot (x) cos^2 (x) dx= int cot(x)dx -int cot(x)sin^2(x)dx#
Use the identity #cot(x) = cos(x)/sin(x)#
#int cot (x) cos^2 (x) dx= int cot(x)dx -int cos(x)/sin(x)sin^2(x)dx#
Cancel #sin(x)/sin(x)#:
#int cot (x) cos^2 (x) dx= int cot(x)dx -int cos(x)sin(x)dx#
Use the identity #2cos(x)sin(x) = sin(2x)#
#int cot (x) cos^2 (x) dx= int cot(x)dx - 1/2int sin(2x)dx#
You can find these integrals in any list of integrals:
#int cot (x) cos^2 (x) dx= ln|sin(x)|+ 1/4cos(2x)+C#
