Evaluate definite and indefinite integrals online with step-by-step working.
An integral calculator evaluates both indefinite integrals (antiderivatives) and definite integrals (areas under curves) of functions. Integration is the inverse of differentiation by the Fundamental Theorem of Calculus, and is used to compute areas, volumes, total accumulated quantities (distance from velocity, total revenue from marginal revenue, etc.), probabilities, and centres of mass.
Many integrals have no closed form (e.g. ∫e^(−x²)dx) and must be evaluated numerically. The calculator uses symbolic integration where possible, and numerical methods (Simpson's rule, Gauss quadrature) as a fallback for definite integrals.
\int_a^b f(x) \, dx = F(b) - F(a) \quad \text{where } F' = f
Inputs: Evaluate ∫₀¹ x² dx
An indefinite integral ∫f(x)dx returns the family of all antiderivatives of f, expressed as F(x) + C where C is an arbitrary constant. A definite integral ∫ₐᵇf(x)dx returns a single number — the net signed area between the function and the x-axis from x = a to x = b. The Fundamental Theorem links them: the definite integral equals F(b) − F(a).
When the integrand is a product of two functions where one becomes simpler when differentiated and the other stays manageable when integrated. Common cases: ∫x·eˣ dx, ∫x·cos(x) dx, ∫ln(x) dx, ∫x²·sin(x) dx. The technique repeatedly applies the formula ∫u dv = uv − ∫v du.
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