Integral Calculator

Evaluate a definite integral between two bounds with Simpson's rule (1000 subintervals), or switch to indefinite to find the antiderivative F(x) + C. Handles most smooth functions.

Calculus Integration Numerical Methods

Inputs

Input examples

Type of integral

Between two curves

The area enclosed by f(x) and a second function g(x): ∫ |f − g| dx

f(x) — e.g. sin(x), x^2, e^x

Lower bound a

Upper bound b

What is an Integral?

The definite integral ∫_a^b f(x) dx gives the net signed area between the curve f(x) and the x-axis from a to b. Areas above the x-axis are positive; areas below are negative.

The indefinite integral ∫ f(x) dx = F(x) + C is the reverse of differentiation: a function F whose derivative is f. The two are linked by the fundamental theorem of calculus: ∫_a^b f(x) dx = F(b) − F(a).

This calculator uses Simpson's rule — a numerical method that approximates the area using parabolic segments. With 1000 subintervals it is accurate to at least 8 significant figures for smooth functions.

Result > 0more area above x-axis

Result < 0more area below x-axis

Result = 0areas cancel out

a = bresult is always 0

Live Graph

f(x)

Function graph

Enter a function to see its graph.
The shaded area appears after Calculate.

f(x)

∫_a^b f(x) dx — shaded area

Result

Enter a function and bounds above

Step-by-Step Solution

The steps appear here after you calculate.

Common Antiderivatives

+ C omitted

f(x)	∫ f(x) dx

Integration Methods

Integration by Parts

Use when integrating a product of two functions. Choose u to differentiate and dv to integrate. Tip: pick u as the function that simplifies when differentiated (LIATE order: Logarithm, Inverse trig, Algebraic, Trig, Exponential).

Example:

U-Substitution

Use when the integrand contains a composite function and its derivative. Let u = inner function, then du = u' dx and substitute.

Example:

Sources & Verification

References

J. Stewart, Calculus: Early Transcendentals, 8th ed., Cengage, 2016 — Ch. 5 (Integrals) and Ch. 7 (Techniques of Integration).
M. Abramowitz & I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover, 1964 — Ch. 7 (Error Function and Fresnel Integrals) and Ch. 25 (Numerical Interpolation, Differentiation, and Integration).