# How do you integrate  ((ln(x))^7)/x dx?

Mar 14, 2018

$\frac{1}{8} {\left(\ln \left(x\right)\right)}^{8} + C$

#### Explanation:

We have: $\int$ $\frac{{\left(\ln \left(x\right)\right)}^{7}}{x}$ $\mathrm{dx}$

$= \int$ ${\left(\ln \left(x\right)\right)}^{7} \cdot \frac{1}{x}$ $\mathrm{dx}$

Let $u = \ln \left(x\right) R i g h t a r r o w \mathrm{du} = \frac{1}{x}$ $\mathrm{dx}$:

$= \int$ ${u}^{7}$ $\mathrm{du}$

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

$= \frac{1}{8} {u}^{8} + C$

Replace $u$ with $\ln \left(x\right)$:

$= \frac{1}{8} {\left(\ln \left(x\right)\right)}^{8} + C$