Find a verified symbolic antiderivative or evaluate a finite definite integral. The calculator shows the integration rules used, applies adaptive numerical integration when needed, and graphs the integrand with signed area—all locally in your browser.
Use ^ for powers. Functions include sin, cos, tan, sec, csc, cot, asin, acos, atan, sinh, cosh, tanh, sech, exp, ln, log, sqrt, and abs.
Keyboard shortcut: Ctrl/⌘ + Enter.
Integration steps will appear here.
Integrate a function to draw its curve. Definite mode shades the interval between the curve and horizontal axis.
4x^3-5x+7, 2x*cos(x^2), or exp(-x^2).Multiplication may be explicit or implicit. Function names require parentheses, and trigonometric inputs use radians.
Polynomials, linear powers, reciprocal-linear forms, common trig and hyperbolic functions, exponentials, logarithms, inverse-trig primitives, recognizable u-substitution patterns, and polynomial products handled by repeated integration by parts produce verified formulas.
A continuous definite integral can still be evaluated when the engine has no implemented elementary primitive. Refined Simpson estimates continue until the selected tolerance is met.
When an antiderivative F is available, the definite integral is evaluated as F(b)−F(a) and cross-checked numerically.
Area above the horizontal axis contributes positively and area below it contributes negatively. The graph uses separate shading for both.
∫(4x³−5x+7)dx
Increase each exponent and divide: x⁴−(5/2)x²+7x+C.
∫2x cos(x²)dx
Let u=x², so du=2x dx. The result is sin(x²)+C.
∫₀^π sin(x)dx
An antiderivative is −cos(x), so −cos(π)−[−cos(0)]=2.
∫₀¹ exp(−x²)dx
This non-elementary antiderivative is evaluated numerically as approximately 0.7468241328.
An indefinite integral returns a family of antiderivatives with + C. A definite integral returns accumulated signed area between two bounds.
Indefinite mode reports that no supported elementary form was found. With finite bounds, definite mode uses adaptive Simpson integration and labels the answer numerical.
It differentiates the candidate antiderivative and compares the result with the original integrand at representative valid points.
The function may be undefined, non-real or apparently singular inside the interval. Split the interval only when the corresponding improper integrals are mathematically valid.
It represents signed area. Portions below the axis subtract from portions above it.
Yes. Standard calculus integration rules assume radian measure.
No. Symbolic matching, differentiation checks, quadrature and graphing run locally on your device.