Question

How do you integrate `int lnx dx` using integration by parts?

Answer 1

`int lnxdx = x ln x - x + C`

Explanation:

The formula for integration by parts, just for reference:

`int u dv = uv - int v du`

It's not all that obvious what we should set `u` and `dv` as, since we only have one function here.

However, let's set `u = ln x`. Then `du = 1/x dx`.

The real trick is to set `dv = dx`, making `v = x`. This will allow us to get rid of the `ln` and simply be left with a polynomial to integrate.

Plugging into our formula we have

`int ln x dx = x ln x - int x/x dx`

`x/x` is just one:

`int ln x dx = x ln x - int dx`

And we know `int dx` turns out to be `x`. Don't forget the constant of integration:

`int ln x dx = x ln x - x + C`