Education
Integral Calculator
Type a polynomial in x and we'll integrate it term by term using the power rule. Choose indefinite for the antiderivative or definite to evaluate between two bounds — with a numeric Simpson's rule cross-check.
Polynomials in x only — terms like 3x^2, -2x, and 5. No functions like sin or log.
Mode
∫ f(x) dx =
x^3 +x^2 +x + C
Power rule applied term by term, plus constant of integration
Polynomial integrals only — power rule, term by term. The numeric row uses Simpson's rule on 1,000 subintervals as a sanity check. For trig, exponential, logarithmic, or composite integrands, use a full CAS.
Examples
∫ 3x² + 2x + 1 dx
x³ + x² + x + C
∫ x² dx, from 0 to 2
8/3 ≈ 2.667
∫ 4x dx, from 1 to 3
16
∫ 6x² dx, from −1 to 1
4
How it works
Each polynomial term integrates with the power rule. Add a constant of integration C for indefinite integrals, or evaluate the antiderivative at both endpoints for definite integrals.
power rule · ∫ c·xⁿ dx = c·x^(n+1) / (n+1) + C
definite · ∫ from a to b of f(x) dx = F(b) − F(a)
For a sanity check, the definite mode also runs Simpson's rule with 1,000 subintervals — for polynomials the two should match.
Supported: polynomials in x · ^ for exponents · constants · negative coefficients. Unsupported: 1/x, sin, cos, ln, e^x, products, quotients, substitution.
Polynomials only. For 1/x, trig, exponential, logarithmic, or composite integrands, use a full computer-algebra system. We deliberately keep the scope narrow so every step is auditable.
Frequently asked questions
Polynomials in x only — terms like 3x^2, -2x, and constants. The calculator does not handle 1/x, trig, exponential, logarithmic, or composite integrands. For anything beyond polynomial integration, use a full computer algebra system like Wolfram Alpha or SymPy.
For any term c·xⁿ where n ≠ −1, the antiderivative is c·x^(n+1) / (n+1). We apply this to each term and add the results. The constant of integration C captures the family of all functions whose derivative equals f(x).
By the Fundamental Theorem of Calculus: ∫ from a to b of f(x) dx = F(b) − F(a), where F is any antiderivative of f. We compute F using the power rule, evaluate at both endpoints, and subtract. We also run Simpson's rule numerically (n = 1,000 subintervals) and report it alongside as a sanity check — for polynomials of degree ≤ 3 the two should match exactly to floating-point precision.
The power rule fails when n = −1: dividing by (n + 1) means dividing by zero. The correct antiderivative is ln|x|, which falls outside the polynomial scope of this calculator. We flag the case rather than print a wrong answer.
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