let

Put t=cosx

dt=-sinxdx

Putting

A(1+t)(3+2t)+B(1-t)(3+2t)+C(1+t)(1-t)=1

Now Putting 1+t=0

t=-1

A(0)+B(2)(3-2)+C(0)=1

Now Putting 1-t=0

t=1

A(2)(5)+B(0)+C(0)=1

Now Putting 3+2t=0