Calculate Improper Integral Online

Improper integrals extend the concept of integration to functions with infinite limits or singularities. This guide explains how to evaluate them, provides examples, and includes an online calculator for quick results.

What is an Improper Integral?

An improper integral is an integral where either the integrand has an infinite discontinuity (vertical asymptote) or the interval of integration is infinite. These integrals are called "improper" because they don't fit the standard definition of an integral over a finite interval with a finite integrand.

An improper integral of the form ∫_a^∞ f(x) dx is defined as:

lim_b→∞ ∫_a^b f(x) dx

Similarly, for an infinite discontinuity at c:

∫_a^b f(x) dx = lim_t→c⁻ ∫_a^t f(x) dx + lim_t→c⁺ ∫_t^b f(x) dx

Types of Improper Integrals

There are three main types of improper integrals:

Type 1: Infinite interval of integration (e.g., ∫₁^∞ 1/x² dx)
Type 2: Infinite discontinuity in the integrand (e.g., ∫₀¹ 1/√x dx)
Type 3: Both infinite interval and discontinuity (e.g., ∫₁^∞ 1/x dx)

An integral converges if the limit exists and is finite; otherwise, it diverges.

How to Calculate Improper Integrals

To evaluate an improper integral:

Identify the type of improper integral
Set up the limit(s) that define the improper integral
Evaluate the limit(s)
Determine if the integral converges or diverges

Remember: If any part of the improper integral diverges, the entire integral diverges.

Examples of Improper Integrals

Example 1: ∫₁^∞ 1/x² dx

lim_b→∞ ∫₁^b x⁻² dx = lim_b→∞ [-x⁻¹]₁^b = lim_b→∞ (-1/b + 1) = 1

This integral converges to 1.

Example 2: ∫₀¹ 1/√x dx

lim_t→0⁺ ∫_t¹ x⁻¹/² dx = lim_t→0⁺ [2√x]_t¹ = lim_t→0⁺ (2 - 2√t) = 2

This integral converges to 2.

Common Pitfalls

Assuming all improper integrals converge
Incorrectly setting up limits for Type 2 integrals
Forgetting to check both limits for Type 3 integrals
Miscounting the number of infinite limits in a multiple-infinite integral

Frequently Asked Questions

What's the difference between a proper and improper integral?

A proper integral has finite limits and a finite integrand. An improper integral has at least one infinite limit or an infinite discontinuity in the integrand.

How do you know if an improper integral converges?

An improper integral converges if the limit exists and is finite. If the limit is infinite or doesn't exist, the integral diverges.

Can you integrate a function with a vertical asymptote?

Yes, but you must evaluate the integral as a limit approaching the point of discontinuity.