Definite Integral (Polynomial)

∫ₐᵇ p(x) dx using power rule for antiderivatives.

Inputs

Result

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How to use this calculator

  • Enter coefficients up to x³ + bounds a, b.

About this calculator

Definite integral = signed area under curve from a to b. Fundamental theorem of calculus: ∫ₐᵇ f(x) dx = F(b) − F(a) where F'(x) = f(x). Power rule for antiderivative: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. For polynomials, integration is mechanical: increment exponent, divide by new exponent. Used for area, work, displacement from velocity, average value. Source: NIST DLMF, Wolfram MathWorld - Definite Integral.

Frequently asked

Definite vs. indefinite?+
Definite: number (area between bounds). Indefinite: function (antiderivative + C).
Negative result?+
Means net signed area is negative (curve below x-axis dominates).
Why F(b) − F(a)?+
Fundamental theorem of calculus. Connects derivatives and integrals.
Numerical integration?+
For non-polynomial: Simpson's rule, trapezoidal, Romberg. See riemann-sum.
Improper integral?+
Bounds at infinity or discontinuity. Take limit. Calc handles finite polynomials only.

Related calculators

Polynomial Derivative
d/dx(aₙxⁿ + … + a₁x + a₀) = naₙxⁿ⁻¹ + … + a₁. Power rule.
Limit Evaluator (Polynomial Ratio)
lim x→a of polynomial ratios (basic substitution + L'Hopital for 0/0).
Taylor Series Approximation
Approximate sin/cos/eˣ/ln(1+x) via Taylor polynomial of degree n.
Riemann Sum (Numerical Integration)
Approximate ∫ₐᵇ f(x) dx using N rectangles (left, right, midpoint).
Newton's Method Root Finder
xₙ₊₁ = xₙ − f(xₙ) / f'(xₙ). Iteratively find roots of polynomial.
Partial Fraction Decomposition (2 distinct linear)
P(x) / [(x−a)(x−b)] = A/(x−a) + B/(x−b). Distinct linear factors only.

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