Integration Calculator: Definite (numeric) & Indefinite (antiderivative)
Enter f(x) and [a,b]; compute the definite integral by Simpson's rule and show the analytic antiderivative for common functions, with substitution and by-parts notes.
Input Data
Results
At a glance:The definite integral ∫ₐᵇ f(x)dx is the net area under the curve; the indefinite integral is the antiderivative F(x)+C with F'(x)=f(x). Techniques: substitution (u=g(x), du=g'(x)dx) and integration by parts ∫u dv = uv − ∫v du.
Formula
Definite numeric: Simpson ∫≈(h/3)[f₀+4f₁+2f₂+…+fₙ].
Power: ∫xⁿ dx = x^{n+1}/(n+1) + C (n≠−1).
∫eˣ dx = eˣ + C; ∫1/x dx = ln|x| + C.
Substitution: ∫f(g(x))g'(x)dx = ∫f(u)du.
By parts: ∫u dv = uv − ∫v du.
$$\int_a^b f(x)\,dx = F(b) - F(a)$$$$\int u\,dv = uv - \int v\,du$$How to Use
- Enter f(x) (x, + - * / ^, sin/cos/exp/ln/sqrt, e, pi).
- Enter lower a and upper b (b>a).
- The tool gives the definite integral (Simpson) and the antiderivative for common functions.
- Case studies show substitution and by-parts to compare with numeric results.
Basic antiderivatives
| f(x) | ∫ f(x)dx (+C) | Note |
|---|---|---|
| xⁿ | x^{n+1}/(n+1) | n≠−1 |
| eˣ | eˣ | — |
| 1/x | ln|x| | x≠0 |
| sin x | −cos x | — |
| cos x | sin x | — |
| 1/(1+x²) | atan x | — |
Definite integral is numeric (Simpson). If antiderivative is blank, no elementary form is stored — use the numeric value.
Case Studies
∫₀² x² dx
Antiderivative x³/3; definite = 8/3 − 0 = 2.666667.
Simpson ≈ 2.666667.
Substitution: ∫₀¹ 2x·e^{x²} dx
u=x², du=2x dx → ∫eᵘ du = eᵘ.
= e¹ − e⁰ = e − 1 ≈ 1.718282.
By parts: ∫ x·eˣ dx
u=x, dv=eˣdx → du=dx, v=eˣ.
= x eˣ − ∫eˣ dx = eˣ(x−1) + C.
∫ sin x dx (0 to π)
Antiderivative −cos x; definite = −cos π + cos 0 = 2.
∫ 1/x dx (1 to e)
Antiderivative ln|x|; definite = ln e − ln 1 = 1.
∫₀¹ 1/(1+x²) dx
Antiderivative atan x; definite = atan 1 − atan 0 = π/4 ≈ 0.785398.
FAQ
Definite vs indefinite?
Definite ∫ₐᵇ gives a number (net area); indefinite is the antiderivative family F(x)+C. The Fundamental Theorem links them: ∫ₐᵇ = F(b)−F(a).
What is substitution?
Set u=g(x), du=g'(x)dx, turning ∫f(g(x))g'(x)dx into ∫f(u)du. Good for products like e^{x²}·2x.
What is integration by parts?
∫u dv = uv − ∫v du, for products like x·eˣ, x·sin x. Pick u so it simplifies on differentiation (LIATE: log > inverse trig > algebraic > trig > exponential).
Is Simpson's rule accurate?
For smooth functions Simpson converges fast (error ∝1/n⁴). This tool uses enough subintervals to match analytic values to several decimals.
Why +C on indefinite?
Antiderivatives are not unique — any constant difference vanishes on differentiation, so we write F(x)+C. The constant cancels in definite integrals.
Antiderivative blank?
This version stores antiderivatives for polynomials and common functions; others (e.g. e^{x²}) have no elementary antiderivative, so use the numeric definite value.
∫1/x near 0?
1/x is undefined at 0 and the integral diverges; the interval must not contain 0.
Which functions are supported?
x, arithmetic, ^, parentheses, sin/cos/tan/exp/ln/sqrt/abs with constants e, pi.
Related Tools
References
Content review: Calculatorism Science Team. Simpson numeric integration and basic antiderivatives verified. Results are for reference only.