Question

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

Answer 1

`3/2(ln(x))^2+C`

Explanation:

This is the same as asking:

`intln(x^3)/xdx`

Using the logarithm rule `log(a^b)=blog(a)` we can move the constant out. Then, a good substitution would be `u=ln(x)`, which implies that `du=1/xdx` (see this through taking the derivative of `ln(x)`).

`=3intln(x)/xdx=3intln(x)(1/xdx)=3intudu=3u^2/2=3/2(ln(x))^2+C`

We see that integration by parts isn't even necessary!