Improper Integral Calculator

∫

(

Function:

)

w.r.t:

This improper integral calculator computes the values of improper integrals. Serving as a converge or diverge improper integral calculator it also determines whether the given function is convergent or divergent, eliminating potential human errors. It provides a clear, detailed solution to help you solve complex integral problems with confidence.

What is an Improper Integral?

In calculus, an improper integral is an extension of the definite integral. It is especially used when the limits are infinite or when the function being integrated has a discontinuity within the interval. improper integrals address situations where the calculated area extends to infinity or involves a function that is undefined at certain points. An improper integral represents the reverse process of differentiation.

Types of Improper Integrals:

Type 1 (Infinite Limits of Integration):

In type One improper integrals, one or both limits of integration are infinite. Consider a function f(x) defined on the interval [a, ∞). To evaluate the integral of f(x) over this infinite interval, we use a limit:

∫_a^∞ f(x) dx = lim_N→∞ ∫_a^N f(x) dx

It means that we integrate the function f(x) from “a” to an infinite value “n” and after that find the limit when the value of “n” approaches infinity.

Let us suppose that we have a function f(x) which is defined for the interval [a, ∞). Now, if we consider to integrate over a finite domain, the limits become:

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

If the function is defined for the interval (-∞, b], then the integral becomes:

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

improper integral

It should be remembered that if the limits are finite and result in a number, the improper integral is convergent. But if limits are not a number, then the given integral is divergent.

Now, let us discuss the case in which our improper integral has two infinite limits. In this situation, we choose an arbitrary point and break the integral at that particular point. After doing so, we get two integrals having one of the two limits as infinite.

∫-∞∞ f(x) dx = ∫-∞c f(x) dx + ∫c∞ f(x) dx.

improper integral

You can easily evaluate these integrals, now with single infinite limits, by using our type 1 improper integral calculator.

Type 2 (Integrals with Discontinuities):

These integrals have undefined integrands at one or more points of integration. Let f(x) be a function that is discontinuous at x = b and is continuous in the interval [a, b).

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

improper integral type 3

Like above, we consider that our function is continuous at the interval (a, b] and discontinuous at x = a:

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

improper integral type 3

Now if the function is continuous at the interval (a, c] ⋃ (c, b] with a discontinuity at x = c.

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

For quick solutions to these types of integrals, especially those with discontinuities, consider using a type 2 improper integral calculator.

improper integral type 3

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 of Improper Integral

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

Step 2: Rewrite Using Limits

Replace infinite limits or discontinuities with variables, and express them as a limit. If the limit of the integral is infinite:

Use the variables (e.g., 'n' or 't') at the place of the infinite limit
When the variables go to infinity, consider the integral as the limit of the definite integral

If the integral has a discontinuity:

Place the variable (e.g., 'τ') at the point of discontinuity
When the variables go to discontinuity, write the integral as the limit of the definite integral

Step 3: Integrate

Compute the definite integral for the variable.

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, it converges; otherwise, it diverges.

Step 6: Split If Necessary

For discontinuities within the interval or double infinite limits, split the integral and repeat steps 2-5 for each part. Keep in mind splitting is necessary because limits cannot be evaluated across points of discontinuity or across both infinite limits at the same time.

If you want to check your work or quickly evaluate improper integrals, consider using our online improper integral calculator.

Now, let's walk through a couple of manual examples to help you see how it's done.

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

FAQ’s:

How Do You Identify an Improper Integral?

An integral is improper if it has:

Infinite integration limits
Discontinuities within the interval

How Can You Tell If an Improper Integral Converges?

Evaluate the limit:

Finite limit → convergent
Infinite/undefined limit → divergent

Can Improper Integrals Be Split?

Yes, splitting is required for internal discontinuities or when both limits are infinite.

Why Are Improper Integrals Important?

They allow evaluation of integrals over infinite intervals or with discontinuous functions, which standard definite integrals cannot handle.

Is Zero Convergent or Divergent?

Zero itself is neither; convergence/divergence applies to limits of integrals, series, or sequences.

References:

Wikipedia: Improper Integral

Khan Academy: Divergent Improper Integrals