Question

How do you integrate `int ln(sint)cost` by integration by parts method?

Answer 1

`intln(sint)costdt=sint(ln(sint)-1)+C`

Explanation:

First let `w=sint`, implying that `dw=costdt`. This will help simplify the problem before we attempt integration by parts. Through this substitution:

`I=intln(sint)costdt=intln(w)dw`

Now we should apply integration by parts. Let:

`{(u=ln(w)" "=>" "du=1/wdw),(dv=dw" "=>" "v=w):}`

Thus:

`I=wln(w)-intw1/wdw=wln(w)-intdw`

Integrating and factoring:

`I=wln(w)-w=w(ln(w)-1)`

From `w=sint`:

`I=sint(ln(sint)-1)+C`