Question

How do you integrate `int ln(x+3)` by integration by parts method?

Answer 1

First use `u`-substitution, then use integration by parts.

Explanation:

First, use `u` substitution, where `u=x+3`, `du=dx`, and rewrite as

`intln(u)du`

Using the integration by parts method, the integral can be rewritten in the form `uv-intvdu`.

To do this, you must choose some term in the original integral to set equal to `u` and one to set equal to `dv`. That which you set equal to `u` will be derived and that which you set equal to `dv` will be "anti-derived."

I would set `u=ln(u)` and `dv=1`. This gives `du=1/(u)du` and `v=u`. By the above form,

`u ln(u)-intu/udu`

`=>u ln(u)-int1du`

`=>u ln(u)-u+C`

`=>(x+3)ln(x+3)-(x+3)+C`

Hope that helps!