# How do you integrate 4 /((x + 2)(x + 3))?

##### 1 Answer
Mar 20, 2018

The answer is $= 4 \left(\ln \left(| x + 2 |\right) - \ln \left(| x + 3 |\right)\right) + C$

#### Explanation:

Perform the decomposition into partial fractions

$\frac{4}{\left(x + 2\right) \left(x + 3\right)} = \frac{A}{x + 2} + \frac{B}{x + 3} = \frac{A \left(x + 3\right) + B \left(x + 2\right)}{\left(x + 2\right) \left(x + 3\right)}$

The denominators are the same, compare the numerators

$4 = A \left(x + 3\right) + B \left(x + 2\right)$

Let $x = - 2$, $\implies$, $4 = A$

Let $x = - 3$, $\implies$, $4 = - B$

Therefore,

$\frac{4}{\left(x + 2\right) \left(x + 3\right)} = \frac{4}{x + 2} + \frac{- 4}{x + 3}$

$\int \frac{4 \mathrm{dx}}{\left(x + 2\right) \left(x + 3\right)} = \int \frac{4 \mathrm{dx}}{x + 2} + \int \frac{- 4 \mathrm{dx}}{x + 3}$

$= 4 \ln \left(| x + 2 |\right) - 4 \ln \left(| x + 3 |\right) + C$