integration of (sin(x))^5/(cos(x))^3

Question

details of this integration and the answer

Answer 1

=((sin x)^4 (sin x))/(cos x)^3

= (((1- (cos x)^2)^2 (sin x))/(cos x)^3

 

=(1  -    2(cos x)^2   +   (cos x)^4)(sin x)/(cos x)^3

 

=(sin x)/(cos x)^3  -  2(sin x)/(cos x) +  (sin x)(cos x)

 

Integrate, using the substitution u = cos x

 

= (1/2)(cos x)^(-2)  +  2 ln (cos x)  - (1/2)(cos x)^2