Question

What is `int (ln(sinx))*cosxdx`?

Answer 1

`sin(x)ln(sin(x))-sin(x)+C`
or
`sin(x)(ln(sin(x))-1)+C`

Explanation:

`int(ln(sin(x))*cosxdx`

let `s = sin(x)` (normally I would use "u", but we will use it later)
then `ds = cos(x)dx`

now we have
`intln(s)*ds`

To integrate this we use integration by parts
`int udv = uv - int vdu`

let `u = ln(s)` let `dv = ds`
so `du = 1/sds` and `v=s`

the integral can now be rewritten as
`ln(s)s- int s*1/sds`
`sln(s)- int 1ds`
`sln(s)- s +C`

Now substitute `sin(x)` for `s`

`sin(x)ln(sin(x))-sin(x)+C`
or
`sin(x)(ln(sin(x))-1)+C`