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.
Use the format: ax^n + bx^m + ... + c. Integer or decimal coefficients, non-negative integer exponents. Example: x^3 - 6x + 1
This is the default expression in the calculator. Each term is handled separately.
This is the default expression in the calculator, and each term above was handled separately before being recombined.
| Rule | Form | Result |
|---|---|---|
| Reverse power rule | a * x^n dx | (a / (n+1)) * x^(n+1) + C |
| Constant term | c dx | c * x + C |
| Sum rule | f(x) + g(x) dx | Integrate each term separately |
| Definite integral | from a to b | F(b) - F(a), C cancels out |
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, since differentiating any constant gives zero and integration cannot recover information that was never there to begin with.
For the full theory behind integration, see Wolfram MathWorld's integral page 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.
Integration and differentiation undo each other, and both lean on the same exponent rules.
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.
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.
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.
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.
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.
Marcus Vance spent a decade building data pipelines and testing calculation logic before he started writing about the math behind the tools. On MathCalcTools he checks every formula against the source math, then writes out the steps so a reader can follow the same path a calculator takes.