Partial Fractions
\(\DeclareMathOperator{cosec}{cosec}\)
Integrating Partial Fractions
- You can integrate algebraic fractions by converting them to partial fractions first.
\(\int \frac{x-5}{(x+1)(x-2)} \, dx\)
- \(x-5=A(x-2)+B(x+1)\)
- Let \(x=-1\):
\(-6=-3A\)
so \(A=2\) - Let \(x=2\):
\(-3=B(3)\)
So \(B=-1\) - So \(\int \frac{x-5}{(x+1)(x-2)} \, dx=\int \left( \frac{2}{x+1}-\frac{1}{x-2} \right) \, dx\)
- \(=2\ln |x+1|-\ln |x-2|+c\)
- \(=\ln |\frac{(x+1)^2}{x-2}|+c\)