Introduction
Partial Fraction Decomposition is a rigorous algebraic technique used to break down complex rational functions into simpler fractions that are easier to integrate.
The Decomposition
For a rational function \( \frac{P(x)}{Q(x)} \), where the degree of \( P \) is less than \( Q \), we express it as a sum of simpler fractions. For example:
\[ \frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} \]
By solving for constants \( A \) and \( B \), we can transform a difficult integral into a series of elementary natural log integrations.