∫Definite Integral Calculator
e.g. x^2, sin(x), sqrt(x)
A definite integral computes the net signed area between a function f(x) and the x-axis over an interval [a, b]. It is one of the two fundamental operations of calculus (the other being differentiation) and is the basis for computing areas, volumes, arc lengths, work, probability, and much more. The Fundamental Theorem of Calculus connects integrals to antiderivatives.
- 1Definite integral ∫[a,b] f(x) dx = F(b) − F(a) where F is the antiderivative of f
- 2Our calculator uses Simpson's Rule with 1000 intervals for high-precision numerical approximation
- 3Simpson's Rule: ∫ ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(xₙ)]
- 4Error decreases as O(h⁴) — much faster than the Trapezoidal Rule's O(h²)
- 5Supported functions: x^n, sin, cos, tan, sqrt, ln, exp, pi, e
∫[0,1] x² dx=1/3 ≈ 0.3333Antiderivative of x² is x³/3; F(1)−F(0) = 1/3
∫[0,π] sin(x) dx=2.0000Antiderivative of sin(x) is −cos(x); −cos(π)+cos(0) = 2
∫[1,e] ln(x) dx=1.0000Integration by parts gives [x·ln(x)−x] from 1 to e = 1
| f(x) | ∫f(x)dx | Notes |
|---|---|---|
| xⁿ | xⁿ⁺¹/(n+1) + C | n ≠ −1 |
| 1/x | ln|x| + C | |
| eˣ | eˣ + C | |
| sin(x) | −cos(x) + C | |
| cos(x) | sin(x) + C | |
| 1/√(1−x²) | arcsin(x) + C |
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