Integral Comparison Test Calculator

The Integral Comparison Test is a method used to determine whether an improper integral converges or diverges by comparing it to another integral whose convergence is already known. This calculator helps you apply the test quickly and accurately.

What is the Integral Comparison Test?

The Integral Comparison Test is a technique in calculus used to evaluate the convergence of improper integrals. It works by comparing the integral in question to another integral whose convergence is already known.

There are two versions of the test:

Direct Comparison Test: If the integrand of the unknown integral is always less than or equal to the integrand of a known convergent integral, and the integrals are positive, then the unknown integral converges.
Limit Comparison Test: If the limit of the ratio of the two integrands as x approaches infinity is a positive finite number, then both integrals either converge or diverge together.

Note: The integrals must be positive on the interval of comparison for the Direct Comparison Test to be valid.

How to Use the Calculator

To use the Integral Comparison Test Calculator:

Enter the lower and upper limits of integration (use ∞ for infinity).
Input the integrand function for the integral you want to test.
Enter the integrand function of a known integral for comparison.
Select whether you want to use the Direct Comparison Test or the Limit Comparison Test.
Click "Calculate" to see the result.

The calculator will determine whether the integral converges or diverges based on your inputs.

Formula and Assumptions

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

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

The calculator assumes that the functions are positive on the interval of integration and that the comparison integral's convergence is known.

Worked Example

Let's test the convergence of ∫ from 1 to ∞ of (1/x²) dx using the Direct Comparison Test.

We know that ∫ from 1 to ∞ of (1/x³) dx converges (it equals 0.5). Since 1/x² ≥ 1/x³ for all x ≥ 1, by the Direct Comparison Test, ∫ from 1 to ∞ of (1/x²) dx also converges.

Using the calculator:

Set lower limit: 1
Set upper limit: ∞
Enter test integrand: 1/x²
Enter comparison integrand: 1/x³
Select Direct Comparison Test
Click Calculate

The calculator will confirm that the integral converges.

FAQ

When should I use the Direct Comparison Test versus the Limit Comparison Test?

Use the Direct Comparison Test when the integrand of the unknown integral is always less than or equal to the integrand of a known integral. Use the Limit Comparison Test when the integrands are not directly comparable but their ratio approaches a finite positive limit.

What if the comparison integral diverges?

If the comparison integral diverges, the Direct Comparison Test cannot be used. You would need to find a different comparison integral or use the Limit Comparison Test if applicable.

Can I use the Integral Comparison Test for definite integrals?

No, the Integral Comparison Test is specifically for improper integrals that extend to infinity.