Partial Fractions Before Integration

First make a rational integrand proper. Divide polynomials when the numerator's degree is at least the denominator's degree.

For distinct linear factors, write \(\frac{P(x)}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\), solve for \(A\) and \(B\), then integrate each term using logarithms.

Logarithmic antiderivatives need absolute values: \(\int\frac1{x-a}\,\mathrm dx=\ln|x-a|+C\). Work on an interval that does not cross a pole.

When the denominator is an irreducible quadratic, completing the square may reveal an \(\arctan\) form. Verify the derivative after simplification.