Definite Integral Calculator
Compute the exact definite integral of a common function over an interval [a, b] using its closed-form antiderivative.
Function f(x)
Signed area under the curve from a to b
- 1
Antiderivative at b
F(2) = 2.666667Antiderivative of x² evaluated at the upper limit. - 2
Antiderivative at a
F(0) = 0 - 3
Definite integral F(b) − F(a)
2.666667 − 0 = 2.666667
How does this calculator work?
A definite integral equals F(b) − F(a), where F is the antiderivative of f. Choose a function, enter limits a and b, and this tool applies the exact antiderivative to return the signed area under the curve — no numerical approximation, so the answer is precise within floating-point limits.
Formula
How this is calculated
Pick the integrand f(x) from the list and enter the lower limit a and upper limit b. The calculator looks up the exact antiderivative F(x) for the chosen function — for example x² integrates to x³/3, sin x to −cos x, eˣ to eˣ, ln x to x·ln x − x, 1/x to ln|x|, and √x to (2/3)x^1.5 — then applies the Fundamental Theorem of Calculus.
The Fundamental Theorem states that the definite integral equals the net change of the antiderivative across the interval, F(b) − F(a). This gives the signed area between the curve and the x-axis: regions below the axis subtract from the total. Because exact antiderivatives are used (not numerical approximation), the result carries no discretization error. Angles for sin x and cos x are in radians.
Some integrands have domain restrictions. ln x and √x are only defined for x > 0, so both limits must be positive. The antiderivative ln|x| for 1/x is only valid on an interval that does not contain or cross zero, since the function has a singularity at x = 0. When inputs violate these conditions the calculator reports that valid values are required rather than returning a meaningless number.
Frequently asked questions
Yes. Reversing the limits flips the sign: ∫ₐᵇ f dx = −∫ᵦᵃ f dx, since F(b) − F(a) becomes −(F(a) − F(b)).
Radians. To integrate over a degree range, convert the limits to radians first (multiply by π/180).
The function 1/x is undefined at x = 0 and the integral diverges there, so the antiderivative ln|x| only applies on intervals that stay entirely on one side of zero.
Also known as
TG we-Calculate Editorial Team. (2026). Definite Integral Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/definite-integral-calculator
TG we-Calculate Editorial Team. "Definite Integral Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/definite-integral-calculator.
TG we-Calculate Editorial Team, "Definite Integral Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/definite-integral-calculator
@misc{wecalculate_definite_integral_calculator, title = {Definite Integral Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/definite-integral-calculator}}, year = {2026}, note = {TG we-Calculate} }
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