Integral Calculator

An integral is a mathematical concept representing the area under a curve of a function over an interval. Indefinite integrals give a family of antiderivatives (functions whose derivative equals f(x)), while definite integrals produce a specific numeric value representing the net area between the function and the x-axis over [a, b].

An indefinite integral ∫ f(x) dx returns the antiderivative F(x) + C, where C is the constant of integration. A definite integral ∫ₐᵇ f(x) dx evaluates the antiderivative at the upper and lower bounds using the Fundamental Theorem of Calculus: F(b) − F(a), yielding a single numeric value.

This calculator uses Simpson's Rule, a numerical integration method that approximates the area under a curve by fitting parabolic segments between sample points. The more sub-intervals you choose, the more accurate the result. For standard functions over finite intervals, 1,000 intervals typically gives results accurate to 5–6 decimal places.

You can integrate polynomials (x^2, x^3+2*x), trigonometric functions (sin(x), cos(x), tan(x)), exponential functions (exp(x) or e^x), natural logarithms (log(x)), square roots (sqrt(x)), and combinations thereof. Use * for multiplication and ^ for exponentiation (e.g. 3*x^2 + sin(x)).

The Fundamental Theorem of Calculus links differentiation and integration. It states that if F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x) dx = F(b) − F(a). This means you can evaluate definite integrals by finding the antiderivative and plugging in the bounds, rather than computing infinite sums.

This calculator uses numerical methods (Simpson's Rule) to evaluate integrals, which are inherently approximate. While very accurate for smooth functions with a high number of intervals, functions with singularities (like 1/x near x=0) or rapid oscillations may require care with the chosen bounds. For exact symbolic answers, a computer algebra system (CAS) is needed.

This calculator supports finite numeric bounds only. Improper integrals with infinite limits (e.g. ∫₀^∞) require symbolic computation or special numerical techniques. For such cases, choose a very large finite bound as an approximation when the integrand decays quickly, but be aware this is an estimate.

For indefinite integrals, the antiderivative is not unique — any constant C can be added and the derivative is still f(x). The general antiderivative is written as F(x) + C. When evaluating definite integrals, C cancels out in F(b) − F(a), so it doesn't affect the final numeric result.