# What is the antiderivative of  ln x^(1/3)?

Feb 19, 2016

Use integration by parts and simplify $\ln {x}^{\frac{1}{3}}$ to $\left(\frac{1}{3}\right) \ln x$

#### Explanation:

Let $u = \left(\frac{1}{3}\right) \ln x$ and $\mathrm{dv} = \mathrm{dx}$

Then, $\mathrm{du} = \frac{1}{3 x}$ and $v = x$

Using the integration by parts formula ...

$u v - \int v \mathrm{du}$

$\frac{x}{3} \ln x - \int x \left(\frac{1}{3 x}\right) \mathrm{dx}$

Simplify ...

$\left(\frac{x}{3}\right) \left(\ln x - 1\right) + C$

hope that helped