Trigonometric Substitution Worksheet
Reference: Table of Trigonometric Substitutions
Problems
1.
2.
3.
Partial Fraction Decomposition Summary
Goal
To rewrite a complex rational function as a sum of simpler fractions, which are easier to integrate.
Process
1. Make Proper: If the fraction is improper (degree of P(x) ≥ degree of Q(x)), perform. polynomial long division first.
2. Factor Denominator: Completely factor the denominator Q(x) into linear and irre-ducible quadratic factors.
3. Set Up Decomposition: Based on the factors of Q(x), set up the sum of simpler fractions.
Decomposition Rules
Case 1: Distinct Linear Factors For each factor (ax + b), add a term:
Case 2: Repeated Linear Factors For each factor (ax + b)
k
, add k terms:
Case 3: Distinct Irreducible Quadratic Factors For each factor (ax2 + bx + c), add a term:
Case 4: Repeated Irreducible Quadratic Factors For each factor (ax2 + bx + c)
k
, add k terms:
Selected Problems
Evaluate the following integrals.
1.
2.
3.