What's the difference between definite and indefinite integrals?

An indefinite integral is the family of antiderivatives of a function, written with "+ C". A definite integral has bounds and evaluates to a number — the signed area between the curve and the x-axis from a to b.

What is the power rule for integration?

∫xⁿ dx = xⁿ⁺¹/(n+1) + C for any n ≠ −1: raise the exponent by one and divide by the new exponent. The exception n = −1 gives ∫(1/x) dx = ln|x| + C.

Why do indefinite integrals include + C?

Because differentiation destroys constants: x² + 5 and x² − 3 both differentiate to 2x. The constant C represents the entire family of functions sharing that derivative. In definite integrals the C cancels, which is why bounds remove it.

What is the Fundamental Theorem of Calculus?

It links derivatives and integrals: if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). This converts area problems into antiderivative evaluations and is the basis for how definite integrals are computed.

Can every function be integrated in closed form?

No. Some elementary functions — famously e^(−x²) — have no antiderivative expressible in elementary functions. Their definite integrals are computed numerically (e.g., Simpson's rule) or expressed through special functions like the error function.

Integral Calculator

Compute definite and indefinite integrals step by step. Supports polynomials, trig, exponentials, and substitution — see

Function Settings

Mathematical Expression

Supported: x^n, sin(x), cos(x), e^x, 1/x, constants

Variable

Definite Integral

Integration Rules

• Power Rule: ∫x^n dx = x^(n+1)/(n+1) + C
• Constant Rule: ∫k dx = kx + C
• Sum Rule: ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
• Substitution: ∫f(g(x))g'(x) dx = ∫f(u) du

Indefinite Integral

0.3333333333333333x^3 + C

∫x^2 dx

Solution Steps

1. Power Rule: ∫x^2 dx = 0.3333333333333333x^3 + C

How it works

Integration is the reverse of differentiation: it finds the area under a curve, or accumulates a quantity. A definite integral computes the net area between a function and the x-axis over an interval; the power rule for integration adds one to the exponent and divides by the new exponent.

Power rule for integration

∫ xⁿ dx = x^(n+1) ÷ (n+1) + C        Definite: ∫ₐᵇ f(x) dx = F(b) − F(a)

n: exponent (n ≠ −1)
C: constant of integration (indefinite integrals)

Worked example

Integrate ∫ 2x dx from 0 to 3

Antiderivative of 2x is x²
Evaluate: 3² − 0²

Area = 9.

Good to know

A definite integral gives a number (net area); an indefinite one gives a function plus a constant C.
Area below the x-axis counts as negative in a definite integral.
Integration accumulates — e.g. integrating velocity gives distance traveled.