Question

What's the integral of `int sin(x)tan(x) dx`?

Answer 1

`intsin(x)tan(x)dx = ln|sec(x) + tan(x)| - sin(x) + c`

Explanation:

We have

`intsin(x)tan(x)dx`

Or, in other terms

`intsin^2(x)/cos(x)dx`

Or, since `sin^2(x) = 1 - cos^2(x)`

`int(1 - cos^2(x))/cos(x)dx`

Doing the fraction we have

`int(sec(x) - cos(x))dx`

Break the integral into fractions

`intsec(x)dx - intcos(x)dx`

And those are standard integrals, so evaluate that

`intsin(x)tan(x)dx = ln|sec(x) + tan(x)| - sin(x) + c`