Question

How to integrate `cos(x)ln(sin(x))dx`?

Answer 1

`sin x ln(sin x) - sin x+C`

Explanation:

Substitute `(sinx) = u`. Then `cos x dx = du`

Thus

`int cos(x)ln(sin(x))dx = int ln u du`

We solve this integration by parts (taking #ln u as the first function and 1 as the second function.

`int ln u du = ln u int 1 du - int[(int 1du)times d/(du) ln u]du`
`qquad = u ln u - int (u times 1/u)du = u ln u - int du = u ln u - u +C`

So

`int cos(x)ln(sin(x))dx = sin x ln(sin x) - sin x+C`