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Home/Calculus/Improper Integrals
Chapter 7

Improper Integrals

Extending definite integrals to infinite intervals and unbounded functions—convergence tests, comparison theorems, and the beautiful Gamma function

Infinite Limits

Extend integration to infinity

Singular Points

Handle unbounded integrands

Convergence Tests

Comparison and Cauchy criteria

Gamma Function

Extend factorial to all numbers

Course Sections

CALC-7.1
Definition and Basic Properties
Available

Improper integrals with infinite limits (Type I) and singular points (Type II), basic properties, and Newton-Leibniz formula extension.

4-5 hours5 Topics

Topics Covered

Type I: Infinite limits
Type II: Singular points
Newton-Leibniz extension
Mixed improper integrals
Basic properties
CALC-7.2
Convergence Tests
Available

Comparison tests, p-integral test, Cauchy criterion, and limit comparison for determining convergence.

4-5 hours5 Topics

Topics Covered

p-integral test
Direct comparison test
Limit comparison test
Cauchy criterion
Logarithmic test
CALC-7.3
Absolute and Conditional Convergence
Available

Dirichlet test, Abel test, and the crucial distinction between absolute and conditional convergence.

4-5 hours5 Topics

Topics Covered

Absolute convergence
Conditional convergence
Dirichlet test
Abel test
Oscillating integrals
CALC-7.4
Gamma Function and Applications
Available

The Gamma function, Beta function, Gaussian integral, and applications to probability and physics.

4-5 hours5 Topics

Topics Covered

Gamma function definition
Recurrence relation
Γ(1/2) = √π
Beta function
Applications

Learning Path

1

Definition

5 topics

2

Convergence

5 topics

3

Absolute

5 topics

4

Gamma

5 topics

Prerequisites

Definite Integrals

Riemann sums, Newton-Leibniz formula

Limits

Limit evaluation and L'Hôpital's rule

Integration Techniques

Substitution and integration by parts

Series Convergence (helpful)

Comparison and ratio tests for series

Ready to Master Improper Integrals?

Start with definitions and properties, then progress through convergence tests to the powerful Gamma function