\[\frac{p(x)}{q(x)}\]
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.
\[\begin{equation} \begin{aligned} \int \frac{1}{2(x+1)}+\frac{3}{2(x-5)} dx &=\frac{1}{2}ln(x+1)+\frac{3}{2} ln (x+5)+c \\ &=\frac{1}{2}(ln(x+1)+3 ln(x-5))+c \\ & =\frac{1}{2} ln((x+1)(x-5)^3)+c \end{aligned} \end{equation}\]