Partial fractions

A rational function P(x)/Q(x) is proper when the degree of P is less than the degree of Q. An improper one is first divided out into a polynomial plus a proper fraction.

A proper fraction is split into simpler fractions according to the factors of Q(x), and each piece is integrated separately.

Distinct linear factors: (px + q)/((x − a)(x − b)) = A/(x − a) + B/(x − b), with a ≠ b. Three distinct factors give three such terms.

Repeated factor: (px + q)/(x − a)² = A/(x − a) + B/(x − a)². With (x − a)²(x − b) below, use A/(x − a) + B/(x − a)² + C/(x − b).

Irreducible quadratic factor: (px² + qx + r)/((x − a)(x² + bx + c)) = A/(x − a) + (Bx + C)/(x² + bx + c), where x² + bx + c has no real factors.

The constants come from clearing denominators and then either comparing coefficients or putting convenient values of x (the roots of the factors).

Worked example: 1/((x + 1)(x + 2)) = 1/(x + 1) − 1/(x + 2), so the integral is log|(x + 1)/(x + 2)| + C.

Worked example: (x² + 1)/(x² − 5x + 6) is improper. It equals 1 + (5x − 5)/((x − 2)(x − 3)) = 1 − 5/(x − 2) + 10/(x − 3), so the integral is 10 log|x − 3| − 5 log|x − 2| + x + C.

Shortcut for even functions: in x²/((x² + 1)(x² + 4)) treat x² as y. Then y/((y + 1)(y + 4)) = −(1/3)/(y + 1) + (4/3)/(y + 4), and the integral is −(1/3) tan⁻¹x + (2/3) tan⁻¹(x/2) + C.