Integration by Partial Fractions

Partial Fraction Decomposition

By the end of this lesson, you will be able to decompose rational functions into partial fractions and integrate them.

When to Use Partial Fractions

Use when integrating a proper rational function $\frac{P(x)}{Q(x)}$ where deg($P$) < deg($Q$) and $Q$ factors into linear and/or irreducible quadratic factors.

If deg($P$) ≥ deg($Q$): first perform polynomial long division to get a polynomial plus a proper rational function.

Case 1: Distinct Linear Factors

$\frac{P(x)}{(x-a_1)(x-a_2)\cdots(x-a_n)} = \frac{A_1}{x-a_1} + \frac{A_2}{x-a_2} + \cdots$

Example: $\int\frac{3x+5}{(x-1)(x+2)}, dx$

$\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2}$

Multiply both sides by $(x-1)(x+2)$: $3x+5 = A(x+2)+B(x-1)$

$x=1$: $8 = 3A$ → $A = 8/3$

$x=-2$: $-1 = -3B$ → $B = 1/3$

$\int\frac{3x+5}{(x-1)(x+2)}, dx = \frac{8}{3}\ln|x-1| + \frac{1}{3}\ln|x+2| + C$

Case 2: Repeated Linear Factors

$(x-a)^n$: use $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$

Example: $\frac{x+1}{(x-2)^2} = \frac{A}{x-2} + \frac{B}{(x-2)^2}$

$x+1 = A(x-2)+B$

$x=2$: $3 = B$; expand and match: $A = 1$

$\int\frac{x+1}{(x-2)^2}, dx = \ln|x-2| - \frac{3}{x-2} + C$

Case 3: Irreducible Quadratic Factor

$\frac{Bx+C}{x^2+bx+c}$ where $x^2+bx+c$ doesn't factor over $\mathbb{R}$ (discriminant < 0).

Example: $\frac{2x+1}{x^2+4} = \frac{2x}{x^2+4} + \frac{1}{x^2+4}$

$\int\frac{2x}{x^2+4}, dx = \ln(x^2+4)$ (u-sub with $u = x^2+4$)

$\int\frac{1}{x^2+4}, dx = \frac{1}{2}\arctan\frac{x}{2}$ (using $\int\frac{1}{x^2+a^2}, dx = \frac{1}{a}\arctan\frac{x}{a}$)

Summary

Setup: degree of numerator < degree of denominator; denominator must factor.

Distinct linear: $\frac{A}{x-a}$ for each factor.

Repeated: $\frac{A}{x-a} + \frac{B}{(x-a)^2}+\cdots$

Irreducible quadratic: $\frac{Bx+C}{x^2+bx+c}$; split into arctangent and log parts.

Integration: each piece integrates to a log (linear factor) or arctan (quadratic factor).