# Use partial fractions to find the integral: {eq}\int \frac{x^2 + 12x + 12}{x^3 - 4x} \, \mathrm{d}x {/eq}.

## Question:

Use partial fractions to find the integral: {eq}\int \frac{x^2 + 12x + 12}{x^3 - 4x} \, \mathrm{d}x {/eq}.

## Method of Decomposition into Partial Fractions:

The rational fraction defined by the quotient between a polynomial of second degree and a polynomial of third degree is a proper rational fraction. If the roots of the third degree polynomial are different real numbers {eq}\, x_0 \, {/eq}, {eq}\, x_1 \, {/eq} and {eq}\, x_2 \, {/eq}, by applying the method of decomposition into partial fractions, we get $$\frac{a}{x-x_0} + \frac{b}{x-x_1} + \frac{c}{x-x_2}$$.

\begin{align} f(x) &= \frac{x^2+12x+12}{x^3-4x} \\[0.3cm] \end{align} \\