Integral Calculator
Definite integrals of any function, to 10+ digits, with a plot.
About this tool
Type a function of x and the lower and upper limits (expressions like pi/2 work). The tool returns the integral's value, an error estimate and the number of sub-intervals used, and plots the function with the area shaded.
How numerical integration works
A definite integral ∫ₐᵇ f(x) dx is the signed area between a curve and the x-axis from a to b, with areas below the axis counting as negative. Many integrals have no neat formula (∫ e^(−x²) dx, the bell curve, for one), so this tool computes the number directly. It uses adaptive Gauss–Kronrod quadrature (the G7–K15 rule found in professional numerical libraries): on each interval it evaluates the function at 15 carefully placed points, compares a 7-point and a 15-point estimate to measure its own error, and keeps splitting the intervals where the function is difficult until the result is accurate to about 10 to 12 significant digits. Examples: ∫₀^π sin x dx = 2, ∫₀¹ x² dx = 1/3, ∫₁^e 1/x dx = 1 and ∫₀¹ e^(−x²) dx = 0.746824. Expressions are parsed safely with implicit multiplication, π and e, and functions such as sqrt, sin, cos, tan, ln, log, exp and abs. Honest limits: you get a number, not an antiderivative; limits must be finite; integrable spikes at an endpoint (1/√x at 0) converge slowly, and discontinuities or infinite areas can mislead. The derivative calculator and the graphing calculator are related.
Frequently asked questions
What is a definite integral?
The signed area under a curve between two x-values, with area below the x-axis counted as negative.
How accurate is the result?
Usually 10 to 12 significant digits; the tool shows its own error estimate.
Can it find the antiderivative?
No. It computes the numerical value of a definite integral, not a symbolic formula.
What is the integral of sin x from 0 to π?
Exactly 2.
Is my data sent anywhere?
No. Everything is calculated in your browser, and nothing is stored or sent.