代做math3b Trigonometric Substitution Worksheet代做留学生Matlab编程

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.