# How do you use partial fraction decomposition to decompose the fraction to integrate (x^2+x)/((x+2)(x-1)^2)?

Aug 23, 2015

$\frac{{x}^{2} + x}{\left(x + 2\right) {\left(x - 1\right)}^{2}} = \frac{\frac{2}{9}}{x + 2} + \frac{\frac{7}{9}}{x - 1} + \frac{\frac{2}{3}}{x - 1} ^ 2$:

#### Explanation:

$\frac{{x}^{2} + x}{\left(x + 2\right) {\left(x - 1\right)}^{2}}$

We want

$\frac{A}{x + 2} + \frac{B}{x - 1} + \frac{C}{x - 1} ^ 2 = \frac{{x}^{2} + x}{\left(x + 2\right) {\left(x - 1\right)}^{2}}$:

Clear the denominator to get:

$A \left({x}^{2} - 2 x + 1\right) + B \left({x}^{2} + x - 2\right) + C \left(x + 2\right) = {x}^{2} + x$

So we need to solve the system:

$A + B = 1$
$- 2 A + B + C = 1$
$A - 2 B + 2 C = 0$

Solve to get:

$A = \frac{2}{9}$, $\text{ } B = \frac{7}{9}$, and $\text{ } C = \frac{2}{3}$