Question

What is the antiderivative of `ln(3x)`?

Answer 1

`xln(3x) - x + C`

Explanation:

We can do this in a number of ways. 

First, let's get rid of that three using a log property:
`ln(3x) = ln(3) + ln(x)`
ln(3) is a constant, so its antiderivative will just be `xln(3)` which is fine to deal with later.

For `ln(x)`, we can write the following:
`dy/dx = ln(x) rightarrow dy = ln(x) dx`
Let's use u substitution with `u=ln(x)`, i.e. `du = 1/x dx = e^(-u) dx`. This yields

`dy = u e^u du implies y = int u e^u du`
By integration by parts,
`y = ue^u - int e^u du = ue^u - e^u + C`
Putting back `x`,

`y = xlnx - x + C`

This yields the final antiderivative as
`int\ ln(3x)dx = xln(3) + xln(x) - x + C = xln(3x) - x + C`