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Integral Calculator

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

Function Settings

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

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

x^3/3 + C

∫x^2 dx

Solution Steps

1. Integrate term by term: ∫x^2 dx = x^3/3 + 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
  1. Antiderivative of 2x is x²
  2. 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.

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Frequently Asked Questions

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.