Calculate definite and indefinite integrals with step-by-step solutions. Covers power rule, substitution, and integration by parts. Free online tool.
Integration Type
Quick Examples
Common Integration Rules
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An integral is the reverse operation of differentiation. Given a function f(x), its indefinite integral F(x) satisfies F'(x) = f(x). A definite integral ∫[a,b] f(x) dx computes the net signed area under the curve from x=a to x=b. This calculator supports both indefinite integrals (general antiderivatives) and definite integrals (numeric values), with step-by-step solutions for 20+ function patterns.
Integration is one of the two central operations in calculus. The indefinite integral ∫ f(x) dx = F(x) + C finds a function whose derivative equals f(x), called the antiderivative. The constant C represents a family of solutions. The definite integral ∫[a,b] f(x) dx evaluates to a number — the net signed area between the curve and the x-axis over the interval [a, b].
Fundamental Theorem of Calculus
∫[a,b] f(x) dx = F(b) − F(a)See exactly which rule applies — power rule, trig rule, integration by parts — and how each step is derived. Perfect for checking homework and learning techniques.
Switch between indefinite integrals (antiderivatives with + C) and definite integrals with numeric bounds to get an exact area value.
When no symbolic rule matches, the calculator uses Simpson's Rule with 1000 intervals to approximate definite integrals to high accuracy.
A handy reference table of 10 common integral formulas is always visible, so you can understand patterns and apply them to similar problems.
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An indefinite integral produces a function (the antiderivative) plus a constant C. A definite integral has upper and lower bounds and evaluates to a number — the net signed area under the curve between those bounds. For example, ∫ x dx = x²/2 + C (indefinite), while ∫[0,1] x dx = 1/2 (definite).
An antiderivative of f(x) is any function F(x) such that F'(x) = f(x). Because the derivative of any constant is zero, antiderivatives are not unique — they form a family F(x) + C. The indefinite integral ∫ f(x) dx represents this entire family.
The Fundamental Theorem of Calculus links differentiation and integration. It states that if F is an antiderivative of f, then ∫[a,b] f(x) dx = F(b) − F(a). This means you can evaluate a definite integral by finding any antiderivative and subtracting its values at the bounds.
This calculator supports 22 symbolic patterns including: the constant rule, power rule (xⁿ), reciprocal rule (1/x), exponential rules (eˣ, e^kx), trig rules (sin, cos, tan, sec²), square root, natural logarithm (via integration by parts), arctangent, arcsine, hyperbolic functions (sinh, cosh), integration by parts for x·eˣ, x·sin(x), and x·cos(x), and polynomial sums via the Sum Rule.
For definite integrals, the calculator automatically falls back to numerical integration using Simpson's Rule with 1,000 subintervals, which provides high-accuracy approximations. For indefinite integrals with unrecognized patterns, an error message is shown — try breaking the function into simpler terms or check the quick examples.
Use the caret ^ for exponents: x^2 for x², x^3 for x³. Type e^x for eˣ, e^(2*x) for e^2x. For trig, use sin(x), cos(x), tan(x). For logarithms, use ln(x). For square root, use sqrt(x). For fractions like 1/x, type 1/x. Polynomials can be combined: x^2 + 2*x + 1.
Yes! The calculator recognizes several integration-by-parts patterns directly: ∫ ln(x) dx = x·ln(x) − x + C, ∫ x·eˣ dx = eˣ(x−1) + C, ∫ x·sin(x) dx = sin(x) − x·cos(x) + C, and ∫ x·cos(x) dx = cos(x) + x·sin(x) + C. The step-by-step solution shows the full IBP derivation.