How Integrals Work

If derivatives measure how fast something changes, integrals measure the total accumulation. The integral of a function gives you the area under its curve, the total distance traveled from a velocity function, or the total work done by a force.

Indefinite Integrals

An indefinite integral (antiderivative) is a function whose derivative is the original function. The integral of x² is x³/3 + C, where C is the constant of integration. Key rules: the integral of xⁿ is x^(n+1)/(n+1), the integral of sin(x) is -cos(x), and the integral of e^x is e^x.

Definite Integrals

A definite integral has bounds [a, b] and gives a specific number: the net area under the curve between a and b. The Fundamental Theorem of Calculus connects derivatives and integrals: the definite integral of f from a to b equals F(b) - F(a), where F is any antiderivative of f.

Techniques

Substitution is the integration analog of the chain rule. Integration by parts is the analog of the product rule. Partial fractions decompose rational functions into simpler pieces. Each technique has specific situations where it works best.

Use our Integral Calculator to look up common integrals and verify your work.