Home

The Definite Integral Online Calculator

Definite integral calculator. Enter f(x) and limits a,b to compute the integral by adaptive Simpson's rule with error estimate—for definite-integral checks.

Show more ▾
∫ₐᵇ f(x) dx = F(b) − F(a)
Supports + - * / ^ ( ) and sin cos tan sec cot sqrt exp log abs asin atan pi e; write 2*x, not 2x
Lower limit of integration
Upper limit of integration

📖 Tutorial | The Definite Integral

1. Definition
The definite integral ∫ₐᵇ f(x)dx is the limit of Riemann sums; geometrically it is the signed area under y=f(x) over [a,b]. By the FTC: ∫ₐᵇ f(x)dx = F(b)−F(a).
2. Symbols
SymbolsMeaning
∫Integral sign
a,bLower and upper limits
f(x)Integrand
dxElement
F(x)An antiderivative
3. How it works
  • No antiderivative needed; adaptive Simpson integrates directly;
  • Nodes are refined where the curve bends; error ~1e-10;
  • Great for continuous functions without easy closed forms.
4. Steps
  1. Enter f(x), e.g. x^2;
  2. Enter lower a and upper b;
  3. Click Calculate for ∫ₐᵇ f(x)dx.
5. Example
Example: ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 ≈ 0.333333. Tool: 0.333333.
6. Pitfalls
A definite integral is a number, not a function;
Swapping limits flips the sign: ∫ₐᵇ = −∫ᵇₐ;
Avoid non-integrable singularities inside the interval.

❓ FAQ | The Definite Integral

Definite vs. indefinite integral?
A definite integral is a number; an indefinite integral is the family F(x)+C.
What method does it use?
Adaptive Simpson, refining nodes by curvature; accuracy ~1e-10.
Can it integrate non-integrable functions?
It works for continuous or piecewise-continuous functions without strong singularities; poles/divergence give NaN.
Why decimals?
It is numerical integration, not exact, but the error is tiny.
Which functions?
+−*/^ and sin cos tan sec cot sqrt exp log abs asin atan pi e.