Partial Fraction Decomposition

Part 1 of 7 — The Concept

When to Use

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

Distinct Linear Factors

rac{1}{(x-a)(x-b)} = rac{A}{x-a} + rac{B}{x-b}

Worked Example

int rac{1}{x^2 - 1},dx = int rac{1}{(x-1)(x+1)},dx

rac{1}{(x-1)(x+1)} = rac{A}{x-1} + rac{B}{x+1}

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

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

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

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

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

Example

rac{3x+5}{(x+1)^2} = rac{A}{x+1} + rac{B}{(x+1)^2}

$3x + 5 = A(x+1) + B$

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

Coefficient of $x$: $3 = A$

Repeated Factors 🎯

Key Takeaways — Part 2

1. Repeated factor $(x-a)^n$ needs $n$ terms
2. Each term has increasing powers in the denominator

Partial Fractions

Part 3 of 7 — Integration Practice

The Key Antiderivatives

int rac{A}{x-a},dx = Aln|x-a| + C

Integration Practice 🎯

Key Takeaways — Part 3

Each partial fraction integrates to logs or power rule.

Partial Fractions

Part 4 of 7 — Long Division First

When Degree of Numerator $geq$ Degree of Denominator

You must do polynomial long division first!

rac{x^2 + 1}{x - 1} = x + 1 + rac{2}{x-1}

Then integrate term by term.

Long Division + PFD 🎯

Key Takeaways — Part 4

Always check: is the degree of numerator $geq$ denominator? If so, divide first!

Partial Fractions

Part 5 of 7 — Logistic DE Connection

The Logistic Equation

rac{dP}{dt} = kP(L - P)

Separation of variables:

rac{dP}{P(L-P)} = k,dt

Use partial fractions on the left side!

Logistic Connection 🎯

Key Takeaways — Part 5

Partial fractions are the key technique for solving the logistic DE.

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! ✅