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Integral Calculator

Enter a polynomial like 3x^2 + 2x - 4 to get the indefinite integral with steps. Enable the definite integral option to evaluate between two bounds a and b.

Chris Terry
By Chris Terry, Editor
Updated June 20, 2026

Enter a polynomial

Use the format: ax^n + bx^m + ... + c. Integer or decimal coefficients, non-negative integer exponents. Example: x^3 - 6x + 1

Result

Integral--
Enter a polynomial above to see the steps.

How polynomial integration works

Integration is the reverse of differentiation. For a polynomial, each term is integrated independently using the reverse power rule, then the results are combined and a constant C is added.

Reverse power rule: integral of a * x^n dx = (a / (n+1)) * x^(n+1) + C

Worked example: integrate 3x^2 + 2x - 4

This is the default expression in the calculator. Each term is handled separately.

Term 1: 3x^2 Raise exponent by 1, divide by new exponent.
Integral of 3x^2 dx = (3 / (2+1)) * x^(2+1) = (3/3) * x^3 = x^3

Term 2: 2x Treat as 2 * x^1.
Integral of 2x dx = (2 / (1+1)) * x^2 = (2/2) * x^2 = x^2

Term 3: -4 Treat as -4 * x^0.
Integral of -4 dx = (-4 / (0+1)) * x^1 = -4x

Result: F(x) = x^3 + x^2 - 4x + C

Definite integral example: from x = 0 to x = 3

Use F(x) = x^3 + x^2 - 4x from above.
F(3) = 27 + 9 - 12 = 24
F(0) = 0 + 0 - 0 = 0
Definite integral = F(3) - F(0) = 24 - 0 = 24

For the full theory behind integration, see Wolfram MathWorld on integrals and the MIT OpenCourseWare single-variable calculus course.

This tool covers polynomials. For trig, exponential, or rational integrals, additional techniques such as substitution and integration by parts apply. Paul's Online Math Notes at Lamar University has a thorough introduction.

Related calculators

Also on MathCalcTools: the derivative calculator (the operation integration undoes) and the exponent calculator.

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Good to know

Frequently asked questions

What is an integral?

An integral finds the accumulated area under a curve. The indefinite integral of f(x) is a new function F(x) such that F'(x) = f(x), written as the integral of f(x) dx = F(x) + C, where C is the constant of integration.

What is the reverse power rule for integration?

The reverse power rule states: the integral of a * x^n dx = (a / (n+1)) * x^(n+1) + C, for n not equal to -1. Divide the coefficient by the new exponent and raise the exponent by one.

What is the constant of integration C?

Because differentiation removes any added constant, integration cannot recover it. The + C in the indefinite integral represents that unknown constant. For a definite integral between bounds a and b, C cancels out.

What is a definite integral?

A definite integral evaluates the integral between two bounds a and b. Compute F(b) - F(a) using the antiderivative F(x). The result is a number, not a function, and represents the net area under the curve on that interval.

Can this calculator integrate trig or exponential functions?

This tool integrates polynomial expressions using the reverse power rule. For trig, exponential, or other functions, use a full symbolic solver like Wolfram Alpha or Symbolab.

Chris Terry
About the author
Chris Terry
Editor, Encore Editorial

Editor at Encore Editorial, Chris Terry sets the editorial standards here and turns dense topics into plain English. He has written widely on education, finance, and consumer markets.