# How do you integrate int (x-5)/(x-2)^2 using partial fractions?

Jan 24, 2017

$\int \frac{x - 5}{{\left(x - 2\right)}^{2}} \mathrm{dx} = \ln \left\mid x - 2 \right\mid + \frac{3}{x - 2} + C$

#### Explanation:

We do not really need partial fractions here as we can separate the numerator by writing:

$\int \frac{x - 5}{{\left(x - 2\right)}^{2}} \mathrm{dx} = \int \frac{x - 2 - 3}{{\left(x - 2\right)}^{2}} \mathrm{dx} = \int \frac{\mathrm{dx}}{x - 2} - 3 \int \frac{\mathrm{dx}}{{\left(x - 2\right)}^{2}}$

now, as $d \left(x - 2\right) = \mathrm{dx}$ we can solve the integrals directly

$\int \frac{x - 5}{{\left(x - 2\right)}^{2}} \mathrm{dx} = \ln \left\mid x - 2 \right\mid + \frac{3}{x - 2} + C$