Page 2: Integration Methods and Rational Functions

This page delves into advanced integration techniques, particularly focusing on integration by substitution and parts. The section also introduces the concept of rational function integration.

Definition: Integration by substitution involves transforming an integral through a change of variable t = g(x), where dt = g'(x)dx.

Highlight: Integration by parts formula: ∫udv = uv - ∫vdu, with recommendations for choosing u including logarithmic, polynomial, and trigonometric functions.

Vocabulary: Rational functions are defined as quotients of polynomials P(x)/Q(x), where integration requires division and partial fraction decomposition.

Page 3: Partial Fraction Decomposition

This page details the process of partial fraction decomposition for integrating rational functions, covering different types of roots and their corresponding decomposition methods.

Definition: Partial fraction decomposition breaks down complex rational expressions into simpler fractions that can be integrated individually.

Example: For real simple roots, the decomposition takes the form A/(x-a) + B/(x-b) + C/(x-c).

Highlight: Three main cases are presented:

1. Real and simple roots
2. Real and multiple roots
3. Imaginary simple roots

Vocabulary: Multiplicity refers to how many times a root appears in the denominator polynomial.

Page 1: Indefinite Integration and Basic Primitives

This page introduces the fundamental concepts of indefinite integration and primitive functions. The relationship between derivatives and primitives is established, showing how if F'(x) = f(x), then F is a primitive of function f.

Definition: The indefinite integral represents the collection of all primitive functions of f(x), written as ∫f(x)dx = F(x) + C.

Highlight: Key properties include linearity of integration and the constant multiple rule: ∫kf(x)dx = k∫f(x)dx.

Example: Basic immediate integrals are presented, including:

• Power functions: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
• Exponential functions: ∫eˣdx = eˣ + C
• Trigonometric functions: ∫sin(x)dx = -cos(x) + C