Convergent or Divergent Integral Calculator

Determining whether an integral converges or diverges is a fundamental problem in calculus. This calculator helps you evaluate improper integrals and understand their behavior as the limits approach infinity.

What is Integral Convergence?

An integral is said to converge if its value is finite, and it diverges if the value is infinite. Improper integrals occur when the interval of integration is infinite or the integrand has an infinite discontinuity within the interval.

There are three types of improper integrals:

Type 1: Infinite interval of integration (e.g., ∫ from 1 to ∞ of 1/x² dx)
Type 2: Infinite discontinuity within the interval (e.g., ∫ from 0 to 1 of 1/√x dx)
Type 3: Both infinite interval and discontinuity (e.g., ∫ from 0 to ∞ of e⁻x dx)

Definition: An integral ∫ from a to b of f(x) dx converges if the limit as b approaches ∞ of ∫ from a to b of f(x) dx exists and is finite.

Methods to Determine Convergence

Direct Evaluation

For some integrals, you can directly evaluate the limit to determine convergence. For example:

∫ from 1 to ∞ of 1/x² dx = lim (b→∞) [ -1/x ] from 1 to b = lim (b→∞) [ -1/b + 1 ] = 1

This integral converges to 1.

Comparison Test

The comparison test compares an integral to a known convergent or divergent integral.

If 0 ≤ f(x) ≤ g(x) for x ≥ a, and ∫ from a to ∞ of g(x) dx converges, then ∫ from a to ∞ of f(x) dx converges.

Limit Comparison Test

This test compares the integrand to a known function by taking the limit.

If lim (x→∞) [f(x)/g(x)] = L (0 < L < ∞), and ∫ from a to ∞ of g(x) dx converges, then ∫ from a to ∞ of f(x) dx converges.

Ratio Test

The ratio test is useful for integrals involving exponential or factorial functions.

If lim (x→∞) [ (x+1)f(x+1) / f(x) ] = L < 1, then ∫ from 1 to ∞ of f(x) dx converges.

Examples of Convergent and Divergent Integrals

Convergent Integral Example

Evaluate ∫ from 1 to ∞ of 1/x² dx

∫ from 1 to ∞ of 1/x² dx = lim (b→∞) [ -1/x ] from 1 to b = lim (b→∞) [ -1/b + 1 ] = 1

This integral converges to 1.

Divergent Integral Example

Evaluate ∫ from 1 to ∞ of 1/x dx

∫ from 1 to ∞ of 1/x dx = lim (b→∞) [ ln|x| ] from 1 to b = lim (b→∞) [ ln(b) - ln(1) ] = ∞

This integral diverges to infinity.

How to Use This Calculator

Enter the integrand function in the input field (e.g., 1/x²)
Specify the lower and upper limits of integration
Select the type of improper integral (Type 1, Type 2, or Type 3)
Click "Calculate" to determine if the integral converges or diverges
Review the result and explanation

Note: This calculator uses numerical methods to approximate the integral value. For exact results, analytical methods should be used.

Frequently Asked Questions

What is the difference between convergent and divergent integrals?: A convergent integral has a finite value, while a divergent integral approaches infinity.
How do I know if an integral is improper?: An integral is improper if it has an infinite interval of integration or an infinite discontinuity within the interval.
What methods can I use to determine convergence?: Common methods include direct evaluation, comparison test, limit comparison test, and ratio test.
Can this calculator handle all types of improper integrals?: This calculator can evaluate Type 1, Type 2, and Type 3 improper integrals using numerical approximation methods.
Is the result always exact?: No, this calculator provides approximate results. For exact values, analytical methods should be used.