Question

How do you find the antiderivative of `int sinx(cosx)^(3/2) dx`?

Answer 1

`-2/5(cosx)^(5/2)+C`

Explanation:

Since we don't want to have to deal with the `3//2` power, let `u=cosx`. This implies that `du=-sinxcolor(white).dx`.

So, we need to modify our integral just a little:

`intsinx(cosx)^(3/2)dx=-int(cosx)^(3/2)(-sinxcolor(white).dx)=-intu^(3/2)color(white).du`

Integrate this using the rule `intu^ncolor(white).du=u^(n+1)/(n+1)`, where `n!=-1`.

`=-u^(5/2)/(5/2)=-2/5u^(5/2)=-2/5(cosx)^(5/2)+C`