Integral Calculator

Integral Calculator: Definite & Indefinite Integrals

Our Integral Calculator computes both indefinite integrals (antiderivatives with + C) and definite integrals (the exact numeric area under a curve between two bounds). Integration is the inverse of differentiation and forms the backbone of advanced calculus and physics.

Indefinite vs Definite Integrals

An indefinite integral produces a family of functions, always appended with + C. A definite integral ∫[a→b] f(x)dx evaluates the net signed area between the function and the x-axis. Our calculator uses Simpson's Rule — a high-precision numerical algorithm — to evaluate definite integrals to six decimal places.

Common Integration Rules

• Power Rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
• Trig: ∫sin x dx = −cos x + C
• Exponential: ∫e^x dx = e^x + C
• Reciprocal: ∫(1/x) dx = ln|x| + C

Real-World Applications

Integrals model total distance, electric charge, fluid volume, and probability. In economics, the consumer surplus is the integral of demand above market price. Every physics simulation computing trajectory or energy uses integral calculus at its core.