# How do you integrate x/(x-6)?

##### 2 Answers

It is

$\int \left(\frac{x}{x - 6}\right) \mathrm{dx} = \int \left(1 + \frac{6}{x - 6}\right) \mathrm{dx} = x + 6 \cdot \ln \left\mid x - 6 \right\mid + c$

Feb 28, 2017

$\int \frac{x}{x - 6} \mathrm{dx} = x + 6 \ln \left\mid x - 6 \right\mid + C$

#### Explanation:

Complete the numerator to separate the function in a polynomial plus a proper rational function:

$\frac{x}{x - 6} = \frac{x - 6 + 6}{x - 6} = 1 + \frac{6}{x - 6}$

We can now integrate the two terms separately, using linearity:

$\int \frac{x}{x - 6} \mathrm{dx} = \int \mathrm{dx} + 6 \int \frac{\mathrm{dx}}{x - 6}$

$\int \frac{x}{x - 6} \mathrm{dx} = x + 6 \ln \left\mid x - 6 \right\mid + C$