Question

How do you integrate `int cosxln(sinx)` using integration by parts?

Answer 1

It is the same as `int lnt dt`

Explanation:

Let `u = ln(sinx)` and `dv = cosx dx`.

This makes `du = 1/sinx cosx dx` and `v = sinx`

`uv-vdu = sinx ln(sinx) - int sinx(1/sinx cosx) dx`

`= sinx ln(sinx) - int cosx dx`

`= sinx ln(sinx) - sinx +C`.

Note if you know `int ln t dt`

If you know that `int lnt dt = tlnt - t +C`, then you can simply substitue `t = sinx`.