Partial fractions decomposition

Recall that there are three fundamental forms of partial fraction decomposition.
In each of the following forms, let \(P(x)\) be a polynomial such that the resulting fraction on the left-hand-side is proper, i.e., the degree of \(P(x)\) is less than that of the denominator.

1. \( \frac{P(x)}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}\);
2. \( \frac{P(x)}{(ax+b)^2(cx+d)} = \frac{A}{ax+b} + \frac{B}{(ax+b)^2} + \frac{C}{cx+d} \);
3. \(\frac{P(x)}{(ax+b)(x^2+d^2)} = \frac{A}{ax+b} + \frac{Bx+C}{x^2+d^2}\).

The above forms of partial fraction decompositions will prove useful if the integrand involves a rational function.