Question

How do you find the integral `ln(x) x^(3/2) dx`?

Answer 1

To do integration by parts:

`uv = intvdu`

I would pick a `u` that gives me an easy time differentiating, and a `dv` that I would easily be able to integrate multiple times if necessary. If the same integral comes back, or variables aren't going away, either it's cyclic or you should switch your `u` and `dv`.

Let:
`u = lnx`
`du = 1/xdx`
`dv = x^(3/2)dx`
`v = 2/5x^(5/2)`

`=> 2/5x^(5/2)lnx - int2/5(x^(5/2)/x)dx`

Nice! That's not too bad.

`= 2/5x^(5/2)lnx - int2/5x^(3/2)dx`

`= 2/5x^(5/2)lnx - 4/25x^(5/2) + C`

`= 10/25x^(5/2)lnx - 4/25x^(5/2) + C`

`= x^(5/2)[10/25lnx - 4/25] + C`

`= 2/25x^(5/2)[5lnx - 2] + C`