Partial Fractions

Partial Fraction Decomposition

Part 1 of 7 — The Concept

When to Use

For integrals of the form $int \frac{P(x)}{Q(x)},dx$ where $Q$ factors into linear or quadratic terms.

Distinct Linear Factors

$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

Worked Example

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

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

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

$x = 1$: $1 = 2A$, $A = 1/2$

$x = -1$: $1 = -2B$, $B = -1/2$

int left(\frac{1/2}{x-1} - \frac{1/2}{x+1}\right),dx = \frac{1}{2}ln|x-1| - \frac{1}{2}ln|x+1| + C

Partial Fractions 🎯

Key Takeaways — Part 1

1. Factor the denominator first
2. Set up: one fraction per factor
3. Solve for constants using strategic $x$ values

Partial Fraction Decomposition

Part 2 of 7 — Repeated Linear Factors

Repeated Factors

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

Partial Fractions

Part 3 of 7 — Integration Practice

The Key Antiderivatives

$int \frac{A}{x-a},dx = Aln|x-a| + C$

Partial Fractions

Part 4 of 7 — Long Division First

When Degree of Numerator $geq$ Degree of Denominator

You must do polynomial long division first!

$\frac{x^2 + 1}{x - 1} = x + 1 + \frac{2}{x-1}$

Partial Fractions

Part 5 of 7 — Logistic DE Connection

The Logistic Equation

$\frac{dP}{dt} = kP(L - P)$

Partial Fractions

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Partial Fractions — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Partial Fractions — Complete! ✅