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

Dec 3, 2016

$\frac{3}{2} {\left(\ln \left(x\right)\right)}^{2} + C$

#### Explanation:

This is the same as asking:

$\int \ln \frac{{x}^{3}}{x} \mathrm{dx}$

Using the logarithm rule $\log \left({a}^{b}\right) = b \log \left(a\right)$ we can move the constant out. Then, a good substitution would be $u = \ln \left(x\right)$, which implies that $\mathrm{du} = \frac{1}{x} \mathrm{dx}$ (see this through taking the derivative of $\ln \left(x\right)$).

$= 3 \int \ln \frac{x}{x} \mathrm{dx} = 3 \int \ln \left(x\right) \left(\frac{1}{x} \mathrm{dx}\right) = 3 \int u \mathrm{du} = 3 {u}^{2} / 2 = \frac{3}{2} {\left(\ln \left(x\right)\right)}^{2} + C$

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