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Improper Integral Calculator

Input your function and limits to get instant results, including a clear step-by-step breakdown of the convergence/divergence analysis.

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Our improper integral calculator helps you compute improper integrals accurately and efficiently. It can determine whether an integral converges or diverges and provides step-by-step solutions to simplify even the most complex problems.

What is an Improper Integral?

An improper integral is a type of definite integral where the limits of integration are infinite, or the function becomes undefined at certain points within the interval. These integrals allow the analysis of areas over infinite domains or functions with discontinuities. Improper integrals are evaluated using limits, representing an advanced application of calculus.

Types of Improper Integrals:

Type 1 (Infinite Limits of Integration):

Type 1 improper integrals occur when one or both limits of integration extend to infinity. If a function f(x) is defined on [a, ∞), the integral is evaluated as:

∫_a^∞ f(x) dx = lim_N→∞ ∫_a^N f(x) dx

For a function defined on (-∞, b]:

∫_-∞^b f(x) dx = lim_N→-∞ ∫_N^b f(x) dx

If both limits are infinite, split the integral at a convenient point c:

∫_-∞^∞ f(x) dx = ∫_-∞^c f(x) dx + ∫_c^∞ f(x) dx

Each integral is evaluated using limits. Use our type 1 improper integral calculator for quick results.

Type 2 (Integrals with Discontinuities):

Type 2 improper integrals occur when the function is undefined at one or more points in the interval. For example, if f(x) is continuous on [a, b) but undefined at x = b:

∫_a^b f(x) dx = lim_τ→0⁺ ∫_a^b-τ f(x) dx

If the discontinuity occurs at x = a:

∫_a^b f(x) dx = lim_τ→0⁺ ∫_a+τ^b f(x) dx

For a discontinuity at an interior point c:

∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx

Use our type 2 improper integral calculator to solve such problems efficiently.

How To Evaluate An Improper Integral?

Follow these steps to evaluate an improper integral:

Step 1: Identify the Type

Determine if the integral is Type 1 (infinite limits) or Type 2 (discontinuity).

Step 2: Rewrite Using Limits

Replace infinite bounds or points of discontinuity with variables and express the integral as a limit:

For infinite limits, let a variable approach infinity
For discontinuities, let a variable approach the undefined point

Step 3: Integrate

Compute the definite integral for the variable bounds.

Step 4: Apply the Limit

Take the limit of the integral as the variable approaches the bound.

Step 5: Conclude

If the limit is finite, the integral converges; if not, it diverges.

Step 6: Split If Necessary

When internal discontinuities or two infinite limits exist, split the integral and repeat the process for each part.

How To Use The Improper Integral Calculator?

Enter the Function: Type the integrand into the input field
Select Variable: Choose the integration variable
Provide Limits: Enter the lower and upper bounds
Calculate: Click “Calculate” to evaluate
View Output: See whether the integral converges or diverges with step-by-step details