Question

How do you integrate `int x^2 ln 5x dx` using integration by parts?

Answer 1

Start by letting `u=ln(5x)` and `dv=x^2\ dx` to ultimately get `int \ x^2ln(5x)\ dx=1/3 x^3 ln(5x)-x^3/9+C`.

Explanation:

If you let `u=ln(5x)` and `dv=x^2\ dx`, then `du=1/(5x) * 5\ dx=1/x\ dx` and `v=x^3/3`. Therefore

`\int\ x^2ln(5x)\ dx=uv-\int\ v\ du`

`=1/3 x^3 ln(5x)-\int\ x^2/3\ dx=1/3 x^3 ln(5x)-x^3/9+C`.