Question

What is the integral of `int sin^3(x) cos^2(x) dx`?

Answer 1

`(cos^5 x )/5 - (cos^3 )/3 +C`

Explanation:

Write `int sin^3 cos^2 x dx= int sinx (1-cos^2 x) cos^2 x dx`

Now let cos x =t, so that sin dx = -dt. On this substitution, the integral becomes

`-int (1-t^2)(t^2)dt`

`int (t^4-t^2 )dt` = `t^5 /5 -t^3 /3 +C`

No back substitution would have `(cos^5 x )/5 - (cos^3 )/3 +C`