Question

How do you integrate `int sinxln(cosx)` by integration by parts method?

Answer 1

`-cos(x)loge(cos(x))+cos(x) + C`

Explanation:

`d/(dx)(cos(x)log_e(cos(x)))=-sin(x) log_e(cos(x))-sin(x)`

so

`int sin(x) log_e(cos(x))dx =`
`= -int d/(dx)(cos(x)log_e(cos(x)))dx-int sin(x) dx =`
`= -cos(x)loge(cos(x))+cos(x) + C`