## Integration Using Partial Fractions

We can integrate fractions where the denominator is a polynomial using partial fractions, which involves separating the fraction into simpler terms and integrating each one.
Example:
$\int \frac{2}{x^2+3x+2} dx$
.
We solve for
$A$
and
$B$
the expression
$\frac{2}{x^2+3x+2}= \frac{A}{x+1} + \frac{B}{x+2} \rightarrow 2=A(x+2)+B(x+1)$

Let
$x=-2$
then
$2=-B \rightarrow B=-1$

Let
$x=-2$
then
$2=A$

The integral becomes
$\int \frac{2}{x+1} - \frac{2}{x+2} dx$
and the result is
$2 ln(x_1)-2ln(x+2)= 2 ln( \frac{x+1}{x+2} )$
.
In general for each factor in the denominator we must assume a numerator of degree one less than the degree of the numerator, and a repeated factor in the denominator gives rise to terms, with the denominator in each term being powers of the repeated factor.
Example
$\frac{3x-2}{(x^2+1)(x-2)^3}= \frac{Ax+B}{x^2+1}+ \frac{C}{x-2} + \frac{D}{(x-2)^2} + \frac{E}{(x-2)^3}$
.