How to use (3 steps)
- Keep Auto unless you want a specific method; choose degrees/radians once.
- Start from the preset or edit f(x) and [a,b]; tolerance controls when adaptive methods stop.
- Scroll to Result for the value and a short meaning, then review the step log or copy LaTeX/URL.
Results
How it's calculated
Teacher notes
- Adaptive Simpson, Romberg, Gauss-Legendre (2/3/5 nodes), and Monte Carlo are available, with nodes and error estimates recorded in the step log.
- Infinite intervals expand exponentially while monitoring convergence, and a warning appears if divergence is suspected.
How to use this calculator effectively
Use this page when you need a numeric estimate of a definite integral together with method notes, a shaded plot, and a short explanation of what the value means.
Start with the interval and the mode
Enter f(x), then set the lower and upper bounds. Use the standard mode for numeric integration, and switch to the probability view only when your function is a density and you want a cumulative probability over an interval.
How to read the result
The result card gives the integral estimate, the method used, and a short meaning line. The step log explains nodes, subdivisions, or sampling behaviour so you can compare Simpson, Romberg, Gauss-Legendre, and Monte Carlo on the same interval.
Infinite and improper integrals
If you enter ±inf, the calculator treats the problem as an improper integral and checks whether the estimate stabilises as the interval expands. A finite-looking answer should still be interpreted together with the warning and step log.
Common mistakes to avoid
- Using the probability mode for a function that is not a probability density.
- Reading a divergent or slowly converging improper integral as exact because one method returned a number.
- Switching angle units without checking whether your trig function should use radians or degrees.
When to use another page instead
Use the differentiation or graphing pages when you mainly need derivatives, roots, or a visual check before integrating. Use the probability tools when you need a named distribution rather than a custom integrand.
FAQ
Which integration methods are available?
Composite trapezoid, Simpson, adaptive Simpson, Romberg, Gauss-Legendre (2/3/5 nodes), and Monte Carlo are implemented. Each run records nodes, weights, and error estimates in the How it's calculated log.
How do you handle infinite limits or oscillatory integrals?
The integrator expands the interval exponentially while monitoring convergence, and raises an alert when divergence is suspected. Trigonometric functions automatically respect the degree/radian setting.
What is the probability mode for?
Probability mode is for integrating a density over an interval, such as finding the mass between two bounds. It is not a symbolic CDF table, so make sure your function really represents a density on the interval you are using.
Why can different methods disagree slightly?
Different methods sample the interval in different ways, so small differences are normal on difficult functions. If the interval is improper, highly curved, or oscillatory, compare the step log and tighten the tolerance before trusting the last digit.
When should I distrust the numeric answer?
Treat the estimate cautiously when the warning says the integral may diverge, when the method keeps refining without stabilising, or when the result meaning contradicts the graph. In those cases, review the interval and the function before using the number downstream.
How to interpret the output
Numeric estimate first, proof second
This tool is designed for fast numeric checking, not symbolic antiderivatives. Use it to decide whether a value is plausible, then move to a derivation page or notebook if you need a formal proof.
Use the graph and steps together
The shaded plot helps you confirm the interval and sign of the area, while the step log shows how the method sampled the region. If those two views tell different stories, the setup usually needs another look.
Probability interpretation
In probability mode, the result is the area under the density across the chosen interval. A result outside [0,1] is a sign that the function, bounds, or mode do not match a valid probability interpretation.
Improper-integral checklist
For ±inf bounds or endpoint singularities, read the warning text and compare two methods before reporting a final value. Agreement across methods is more informative than a single number on its own.
Suggested next steps
If you want to explain the setup to students, keep the result card visible and walk through the step log line by line. If you need a quick cross-check, copy the share URL and compare another method on the same interval.