Partial Fractions

Reduce degree

ℐ = \int \frac{P (x)}{Q (x)} ⅆ x

P (x) = S (x) Q (x) + R (x) with deg R (x) < deg Q (x)

Thus

\frac{P (x)}{Q (x)} = \frac{S (x) Q (x) + R (x)}{Q (x)} = S (x) + \frac{R (x)}{Q (x)}

ℐ = \int \frac{R (x)}{Q (x)} ⅆ x

\begin{matrix} Q (x) = & {(x - r_{1})}^{m_{1}} \dots {(x - r_{t})}^{m_{t}} \\ . {(x^{2} + u_{1} x + v_{1})}^{n_{1}} \dots {(x^{2} + u_{k} x + v_{k})}^{n_{k}} \end{matrix}

\begin{matrix} \frac{R (x)}{Q (x)} = & \overset{\overset{x - r_{1}}{⏞}}{\frac{A_{1, 1}}{(x - r_{1})} + \frac{A_{1, 2}}{{(x - r_{1})}^{2}} + \dots + \frac{A_{1, m_{1}}}{{(x - r_{1})}^{m_{1}}}} + \dots \\ \dots + \overset{\overset{x - r_{t}}{⏞}}{\frac{A_{t, 1}}{(x - r_{t})} + \frac{A_{t, 2}}{{(x - r_{t})}^{2}} + \dots + \frac{A_{t, m_{t}}}{{(x - r_{t})}^{m_{t}}}} \\ + \overset{\overset{x^{2} + u_{1} x + v_{1}}{⏞}}{\frac{B_{1, 1} x + C_{1, 1}}{x^{2} + u_{1} x + v_{1}} + \frac{B_{1, 2} x + C_{1, 2}}{{(x^{2} + u_{1} x + v_{1})}^{2}} + \dots + \frac{B_{1, n_{1}} x + C_{1, n_{1}}}{{(x^{2} + u_{1} x + v_{1})}^{n_{1}}}} + \dots \\ \dots + \overset{\overset{x^{2} + u_{k} x + v_{k}}{⏞}}{\frac{B_{k, 1} x + C_{k, 1}}{x^{2} + u_{k} x + v_{k}} + \frac{B_{k, 2} x + C_{k, 2}}{{(x^{2} + u_{k} x + v_{k})}^{2}} + \dots + \frac{B_{k, n_{k}} x + C_{k, n_{k}}}{{(x^{2} + u_{k} x + v_{k})}^{n_{k}}}} \end{matrix}