Question

How do you find the integral of `int [cot^5 (x) (sin^4(x) dx]`?

Answer 1

`I=ln|sinx|-sin^2x+sin^4x/4+C`

Explanation:

`int cot^5xsin^4xdx=int (cos^5x)/(sin^5x)sin^4xdx=`

`int (cos^4xcosx)/(sinx)dx=int ((1-sin^2x)^2cosx)/sinx dx=I`

`sinx=t => cosxdx=dt`

`I=int (1-t^2)^2/tdt=int (1-2t^2+t^4)/tdt=int (1/t-2t+t^3)dt`

`I=ln|t|-t^2+t^4/4+C`

`I=ln|sinx|-sin^2x+sin^4x/4+C`