Polynomial Integral Calculator
Enter a polynomial like 3x² and get its antiderivative, each term integrated with the reverse power rule, plus the constant of integration.
How to use this calculator
Type your polynomial using ^ for powers and press Calculate. The calculator integrates term by term with the reverse power rule, shows each term’s transformation, and appends + C, the constant of integration that indefinite integrals always carry.
Scope: indefinite integrals of polynomials, trig functions, and other standard forms; definite integrals with bounds are planned next.
How it works
Integration reverses differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Raise the exponent by one, then divide by the new exponent. Because differentiating kills constants, the antiderivative is only determined up to an additive constant, that is the + C, and dropping it is the classic point-loser.
Worked example
Example: ∫ 3x² dx
Step 1, Raise the exponent: x² → x³.
Step 2, Divide by the new exponent: 3x³ ÷ 3 = x³.
Step 3, Add the constant: x³ + C.
Check: d/dx[x³ + C] = 3x² ✓, differentiation reverses it.
Common mistakes
- Dropping the + C: without it the answer is incomplete, infinitely many functions share the same derivative.
- Dividing by the old exponent: ∫x²dx is x³/3, not x³/2, divide by the new exponent.
- Forgetting ∫k dx = kx: a constant term integrates to itself times x: ∫5 dx = 5x + C.
When to use it
Integrals accumulate: distance from velocity, total cost from marginal cost, area under curves, volumes of revolution, and probabilities from density functions. If the derivative asks “how fast?”, the integral asks “how much in total?”
When the reverse power rule is not enough, integration by parts takes over, see the integral of x ln x, worked example, and integration by parts for x²eˣ in our guides.
Frequently asked questions
∫xⁿ dx = xⁿ⁺¹/(n+1) + C. Increase the exponent by one and divide by the new exponent. Example: ∫x³ dx = x⁴/4 + C.
Because derivatives destroy constants, many functions share one derivative. The + C represents all of them; a definite integral or initial condition pins it down.
Indefinite gives a family of functions (+ C). Definite evaluates between two bounds and gives a number, the net area. This calculator currently does indefinite integrals.
Differentiate it. If you get back the original integrand, the antiderivative is correct, the calculator’s worked example demonstrates this check.
Integral calculator with steps
Our integral calculator shows the full working using the power rule and standard forms. See every step from integrand to antiderivative, verified by calculation.
More integral questions
The integral of 1/x is ln|x| + C, the natural logarithm of the absolute value of x plus the constant of integration. This is the one power-rule exception: for x^n with n = -1, the rule gives division by zero, so the logarithm takes over.
A double integral integrates a function of two variables over a region, written with two integral signs. Evaluate it as two single integrals nested inside each other: integrate with respect to one variable first, treating the other as constant, then integrate the result.