Polynomial Integral Calculator
The antiderivative of a polynomial and its definite integral between two limits — the area under the curve — with the working.
Integrate term by term: a·xⁿ becomes a·xⁿ⁺¹ ÷ (n + 1).
How the polynomial integral calculator works
Integrate term by term: a·xⁿ becomes a·xⁿ⁺¹ ÷ (n + 1). Evaluate the antiderivative at the upper and lower limits and subtract for the definite integral. Enter the coefficients from the highest power down, including zeros for missing terms: 3, 0, −2 is 3x² − 2.
Formula: ∫ a xⁿ dx = a xⁿ⁺¹ ÷ (n + 1); ∫ₐᵇ f = F(b) − F(a)
Worked examples
| Inputs | Definite integral | Note |
|---|---|---|
| 3x² + 2x + 1 from 0 to 2 | 14 | 14 |
| x² from 0 to 3 | 9 | 9 |
| x³ − 4 from −1 to 1 | -8 | −8 |
FAQFrequently asked questions
Why is there a + C?
Differentiation destroys constants, so every antiderivative is defined only up to one. In a definite integral the constant cancels between the two limits, which is why it never appears in the answer.
What if the answer is negative?
The integral is signed: where the curve is below the axis the area counts negative. For the geometric area, split the interval at the roots and add the absolute values.
How do I enter a polynomial with missing terms?
Put a zero in the gap. x³ − 4 is 1, 0, 0, −4; 5x is 5, 0.
Can it integrate other functions?
No — polynomials only. For tabulated data or a function you cannot integrate by hand, use Simpson’s rule with sampled values.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.