Question

What is the antiderivative of `ln x`?

Answer 1

`intlnxdx=xlnx-x+C`

Explanation:

The integral (antiderivative) of `lnx` is an interesting one, because the process to find it is not what you'd expect.

We will be using integration by parts to find `intlnxdx`:
`intudv=uv-intvdu`
Where `u` and `v` are functions of `x`.

Here, we let:
`u=lnx->(du)/dx=1/x->du=1/xdx` and `dv=dx->intdv=intdx->v=x`

Making necessary substitutions into the integration by parts formula, we have:
`intlnxdx=(lnx)(x)-int(x)(1/xdx)`
`->(lnx)(x)-intcancel(x)(1/cancelxdx)`
`=xlnx-int1dx`
`=xlnx-x+C->` (don't forget the constant of integration!)