Integral 0 to Infinity Calculator

Calculator Inputs

Use explicit multiplication like 2*x and x*exp(-x). Supported functions include sin, cos, tan, exp, log, ln, log10, sqrt, and abs.

Function f(x)

Infinity Transform

Tolerance

Minimum Panels

Maximum Panels

Tail Epsilon

Graph Max X

Graph Points

Reset

Example Data Table

Function	Expected Integral	Notes
exp(-x)	1	Classic exponential decay example.
x*exp(-x)	1	Gamma function case with n = 1.
exp(-(x^2))	0.8862269255	Equals √π / 2.
1/(1 + x^2)	1.5707963268	Equals π / 2.
1/(1 + x^4)	1.1107207345	Equals π / (2√2).
sin(x)*exp(-x)	0.5	Damped oscillatory example.

Formula Used

This tool evaluates the improper integral:

I = ∫₀^∞ f(x) dx

Because infinity cannot be sampled directly, the calculator converts the infinite interval into a finite interval.

1) Rational Transform

x = t / (1 - t), dx = dt / (1 - t)²
I = ∫₀¹ f(t / (1 - t)) · 1 / (1 - t)² dt

2) Tangent Transform

x = tan(πt / 2), dx = (π / 2) sec²(πt / 2) dt
I = ∫₀¹ f(tan(πt / 2)) · (π / 2) sec²(πt / 2) dt

3) Adaptive Midpoint Refinement

The finite integral is evaluated with a midpoint rule. The panel count doubles until the difference between consecutive estimates falls below the selected tolerance or the maximum panel limit is reached.

How to Use This Calculator

Enter a function in terms of x.
Use explicit multiplication like 3*x.
Select the rational or tangent infinity transform.
Set tolerance and panel limits for refinement.
Choose graph range and graph point count.
Press Calculate Integral to see the result above the form.
Review the estimated error and convergence message.
Export the current report as CSV or PDF.

Frequently Asked Questions

1) What does this calculator compute?

It estimates improper integrals from zero to infinity for a user-entered function of x. The result is numerical, not symbolic.

2) Can I enter any expression?

You can enter many standard expressions with numbers, x, constants like pi, and functions such as exp, sin, cos, log, sqrt, and abs.

3) Why does the tool use transformations?

A finite numerical rule cannot sample infinity directly. The transformations compress the infinite interval into a finite range that can be integrated safely.

4) What is the difference between the two methods?

The rational transform works well for many decaying functions. The tangent transform can help when the tail behavior fits a trigonometric mapping better.

5) What does estimated error mean?

It is the absolute difference between two successive refined estimates. Smaller values usually indicate a more stable numerical answer.

6) Why might convergence fail?

The function may diverge, oscillate strongly, have a severe singularity, or require tighter settings. Try another transform or higher panel limits.

7) What does the graph show?

The plot shows the function values and the cumulative finite-area build-up over the chosen x range. It helps you inspect decay and tail behavior.

8) Does this replace exact calculus methods?

No. It is a practical numerical tool for estimation, checking, and exploration. Exact symbolic methods remain important when closed forms are available.