Find the indefinite integral `int (tan(x))(ln(cos(x))) dx`

Expert Answers
lemjay eNotes educator| Certified Educator

To solve, apply u-substitution method.

Let,

`u= ln (cosx)`

Then, differentiate u.

`du=1/cosx * (-sinx) dx`

`du=-tanx dx`

`-du=tanx dx`

So, expressing the integrand in terms of u, it becomes:

`int tanx (ln(cosx)) dx`

`= int (ln (cosx)) tanxdx`

`=int u (-du)`

`=- int u du`

`=-u^2/2 + C`

Substituting back x = ln (cosx), it becomes:

`=-(ln(cosx))^2/2+C`

Therefore,` int tanx (ln(cosx)) dx=-(ln(cosx))^2/2+C` .