# How do you integrate (7x + 44)/(x^2 + 10x + 24) using partial fractions?

Jul 6, 2016

$= - \ln \left(x + 6\right) + 8 \ln \left(x + 4\right) + C$

#### Explanation:

$\frac{7 x + 44}{{x}^{2} + 10 x + 24}$

$= \frac{7 x + 44}{\left(x + 6\right) \left(x + 4\right)} q \quad q \quad \star$

$= \frac{A}{x + 6} + \frac{B}{x + 4}$

$= \frac{A \left(x + 4\right) + B \left(x + 6\right)}{\left(x + 6\right) \left(x + 4\right)} q \quad q \quad \square$

so comparing $\star$ to $\square$

x = -4
7(-4) + 44 = B(-4+6), B = 8

x = -6
7(-6) + 44 = A(-6+4), A = -1

so we have

$\int \mathrm{dx} q \quad - \frac{1}{x + 6} + \frac{8}{x + 4}$

$= - \ln \left(x + 6\right) + 8 \ln \left(x + 4\right) + C$