Question

What's the integral of `int sinx * tanxdx`?

Answer 1

`intsin(x)tan(x)dx = intsin^2(x)/cos(x)dx = int(sin^2(x)cos(x))/cos^2(x)dx`

let's `t = sin(x)`

`dt = cos(x)`

`int(t^2)/(1-t^2)dt=int(t^2-1+1)/(1-t^2)dt = -int(1-t^2-1)/(1-t^2)dt`

`=-(intdt-int1/(1-t^2)dt)`

`=-[t]+[arctanh(t)]+C`

Substitute back for `t=sin(x)`

`-[sin(x)]+[arctanh(sin(x))]+C`