Partial fraction decomposition is one of those Calculus II topics that looks manageable in lecture and then falls apart on the exam. The algebra is fussy, the cases multiply, and most textbooks bury the key moves inside dense paragraphs. If you have a test coming up, a problem set due, or a student who needs to get unstuck fast, this guide cuts straight to what matters.

**TLDR: Partial Fraction Decomposition for Integration** covers everything a high school or early college student needs to handle every standard case: polynomial long division to set up improper fractions, full factoring of the denominator, and all four decomposition cases — distinct linear factors, repeated linear factors, irreducible quadratic factors, and repeated irreducible quadratics. Each case is built from the ground up with worked numbers, and three full end-to-end integration problems show how to choose a strategy and carry it through.

This is a focused primer, not a 600-page textbook. It is written for students in AP Calculus BC or a college Calculus II course who need a clear, fast explanation of partial fractions integration techniques — not a review of everything they have ever learned. Every term is defined in plain language, every misconception is named and corrected, and the final section maps where this skill reappears in differential equations, Laplace transforms, and beyond.

Read it in an afternoon. Walk into your exam ready.

• Recognize when a rational function needs partial fraction decomposition before integration
• Perform polynomial long division to reduce improper rational functions
• Set up and solve the correct decomposition for distinct linear, repeated linear, and irreducible quadratic factors
• Integrate each type of partial fraction term, including those producing logarithms and arctangents
• Combine these skills to evaluate definite and indefinite integrals of rational functions end-to-end

1. 1. Why Decompose? The Big Picture
   Motivates the technique by showing why messy rational functions are hard to integrate directly and how splitting them into simple pieces fixes the problem.
2. 2. Setup: Long Division and Factoring the Denominator
   Covers the two prerequisite steps every problem starts with: reducing improper fractions via polynomial long division and fully factoring the denominator over the reals.
3. 3. Case 1 and Case 2: Linear Factors, Distinct and Repeated
   Walks through the decomposition setup and coefficient-solving for denominators with distinct linear factors and with repeated linear factors, with full worked integrals.
4. 4. Case 3 and Case 4: Irreducible Quadratic Factors
   Handles denominators containing irreducible quadratic factors, including repeated ones, and shows how to integrate the resulting Ax+B over quadratic terms using substitution and arctangent.
5. 5. Full Worked Examples End-to-End
   Three complete problems integrating rational functions from start to finish, mixing the cases and showing how to choose strategies and check answers.
6. 6. Where This Shows Up Next
   Brief tour of where partial fractions reappear: differential equations, Laplace transforms, series, and probability, so the reader sees the payoff.