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Home/Calculus/Chapter 5: Indefinite Integration
Chapter 5
17-22 hours total
5 Sections

Indefinite Integration

The indefinite integral, or antiderivative, reverses differentiation. Master the fundamental techniques: substitution, integration by parts, and special methods for rational and irrational functions.

Antiderivatives

Reverse of differentiation

Substitution

Chain rule in reverse

By Parts

Product rule in reverse

Special Methods

Euler & partial fractions

Prerequisites
Chapter 4: Differentiation

Derivatives, chain rule, product rule

Chapter 3: Function Limits

Continuity, intermediate value theorem

Trigonometry

Identities, inverse trig functions

Algebra

Polynomial division, partial fractions

Course Sections

CALC-5.1
3-4 hours

Antiderivatives and Indefinite Integrals

Definition of antiderivative (primitive function), indefinite integral notation, and fundamental properties.

Definition of antiderivative
Indefinite integral notation
Existence of antiderivatives
Uniqueness up to constant
Basic properties
CALC-5.2
3-4 hours

Basic Integration Formulas

Fundamental integration formulas for power, exponential, logarithmic, and trigonometric functions.

Power function integrals
Exponential integrals
Trigonometric integrals
Inverse trigonometric integrals
Hyperbolic function integrals
CALC-5.3
4-5 hours

Integration by Substitution

First and second substitution methods (u-substitution and inverse substitution) with trigonometric and algebraic techniques.

First substitution (u-sub)
Second substitution
Trigonometric substitutions
Algebraic substitutions
Recognizing patterns
CALC-5.4
3-4 hours

Integration by Parts

The integration by parts formula, LIATE rule, reduction formulas, and tabular integration.

Integration by parts formula
LIATE selection rule
Repeated integration by parts
Reduction formulas
Tabular method
CALC-5.5
4-5 hours

Special Function Integration

Integration of rational functions, trigonometric rational functions, and irrational functions with Euler substitutions.

Partial fraction decomposition
Trigonometric rational functions
Universal substitution
Euler substitutions
Binomial integrals
Learning Path
1

Antiderivatives and Indefinite Integrals

Definition of antiderivative (primitive function), indefinite integral notation, and fundamental properties.

2

Basic Integration Formulas

Fundamental integration formulas for power, exponential, logarithmic, and trigonometric functions.

3

Integration by Substitution

First and second substitution methods (u-substitution and inverse substitution) with trigonometric and algebraic techniques.

4

Integration by Parts

The integration by parts formula, LIATE rule, reduction formulas, and tabular integration.

5

Special Function Integration

Integration of rational functions, trigonometric rational functions, and irrational functions with Euler substitutions.