Question

What is the antiderivative of `(sinx + cosx)/tanx`?

Answer 1

`sinx-ln(abs(cotx+cscx))+cosx+C`

Explanation:

We have:

`int(sinx+cosx)/tanxdx=int(sinx+cosx)/(sinx/cosx)dx=int(cosx(sinx+cosx))/sinxdx`

`=intcosxdx+intcos^2x/sinxdx`

The first is a well known integral. For the second term, rewrite `cos^2x` using the Pythagorean Identity:

`=sinx+int(1-sin^2x)/sinxdx=sinx+intcscxdx-intsinxdx`

Again, these are both well-known integrals:

`=sinx-ln(abs(cotx+cscx))+cosx+C`