Question

What is the integral of `ln(x^2)`?

Answer 1

It's `2x ln(x) -2x + C`

By integral I'm assuming it's meant the indefinite integral.

Using the fundamental proprety of logarithms that

`log_c(a^b)=b log_c(a)`

We get

`int ln(x^2) dx = int 2 ln(x) dx = 2 int ln(x) dx`

Integrating by parts, chosing `dv=1` and `u=ln(x)` we get

`int ln (x) dx = x ln(x) - int x/x dx = x ln(x) - int 1 dx = x ln(x) -x + C/2`

(we take the constant to be `C/2` for convinience, and we can do this because the integration constant is taken arbitrarily)

Therefore,

`int ln(x^2) dx = 2x ln(x) -2x + C`