# How do you integrate x/(x+10)?

Apr 9, 2018

$\int \frac{x}{x + 10} \mathrm{dx} = x - 10 \ln | x + 10 | + C$

#### Explanation:

A simple substitution will do.

$u = x + 10 \to x = u - 10$

$\mathrm{du} = \mathrm{dx}$

Rewrite and simplify:

$\int \frac{u - 10}{u} \mathrm{du} = \int \frac{u}{u} \mathrm{du} - 10 \int \frac{\mathrm{du}}{u}$

Integrate:

$\int \mathrm{du} - 10 \int \frac{\mathrm{du}}{u} = u - 10 \ln | u | + C$

Rewrite in terms of $x ,$ yielding

$\int \frac{x}{x + 10} \mathrm{dx} = x + 10 - 10 \ln | x + 10 | + C$

We may absorb the $10$ into $C .$

$\int \frac{x}{x + 10} \mathrm{dx} = x - 10 \ln | x + 10 | + C$