# What is the antiderivative of (ln^6 x)/x?

Jan 31, 2016

$\frac{{\ln}^{7} x}{7} + C$

#### Explanation:

Use substitution.

Let $u = \ln x$, so $\mathrm{du} = \frac{1}{x} \mathrm{dx}$.

Finding the antiderivative is equivalent to finding

$\int \frac{{\ln}^{6} x}{x} \mathrm{dx}$

This can be rewritten as

$= \int {\ln}^{6} x \left(\frac{1}{x}\right) \mathrm{dx}$

Using the substitutions previously defined

$= \int {u}^{6} \mathrm{du}$

This is equal to

$= \frac{1}{7} {u}^{7} + C$

Resubstitute $u$:

$= \frac{{\ln}^{7} x}{7} + C$