Question

Hi I'm really struggleing with this intergral, tried so many ways, just not getting it.

Find the integral of
Cot^2 X cos X dx

Answer 1

Hint:
cot^2(x)=cos^2(x)/sin^2(x)
=(1-sin^2(x))/sin^2(x)
=1/sin^2(x) - 1

cos(x)dx = d(sin(x))

so use the substitution
u=sin(x), then
du=cos(x)dx

cot^2(x) cos(x)dx
=(1/u^2 -1)du
=(-1/u - u) + C
backsubstitute u=sin(x)
I=-1/sin(x)-sin(x)+C