## Integration Using Partial Fractions

To integration an expression of the form
$\frac{p(x)}{q(x)}$
where the degree of
$p(x)$
is less than the degree of
$q(x)$
write the expression as partial; fractions as much as possible and then integrate using other methods.
Example:
$\int \frac{2x-1}{x^2-4x-5} dx$

$\int \frac{2x-1}{x^2-4x-5}=\frac{A}{x+1}+\frac{B}{x-5} \rightarrow 2x-1=A(x-5)+B(x+1)$

$x=-1 \rightarrow 2 \times -1-1=A(-1-5)+B(-1+1)=-6A \rightarrow A=\frac{-3}{-6}=\frac{1}{2}$

$x=5 \rightarrow 2 \times 5-1=A(5-5)+B(5+1)=6B \rightarrow B=\frac{9}{6}=\frac{3}{2}$
.
The integral becomes
$\int \frac{1}{2(x+1)}+\frac{3}{2(x-5)} dx$
.
Now we can integrate.
\begin{aligned} \int \frac{1}{2(x+1)}+\frac{3}{2(x-5)} dx &=\frac{1}{2}ln(x+1)+\frac{3}{2} ln (x+5)+c \\ &=\frac{1}{2}(ln(x+1)+3 ln(x-5))+c \\ & =\frac{1}{2} ln((x+1)(x-5)^3)+c \end{aligned}