Polynomial Integral
Enter a polynomial in x. The calculator finds the antiderivative with the reverse power rule ∫ c·xⁿ dx = c/(n+1)·xⁿ⁺¹ + C. Supply both limits a and b to also evaluate the definite integral as F(b) − F(a).
Exact antiderivatives, and exact areas
Integration is hard in general and trivial for polynomials. Every term has an antiderivative, it is found by one rule, and the rule is the power rule read backwards: raise the exponent, then divide by the exponent you just created.
This calculator applies it term by term. Leave the limits blank and it returns the antiderivative with its constant; supply both and it also evaluates the definite integral.
Because polynomials always integrate in closed form, nothing here is approximated. That is a real contrast with numerical integration, which returns a figure for any integrand but only ever an estimate — here the definite integral is the exact difference of two exact evaluations.
Reporting both the antiderivative and the evaluation keeps the two ideas apart. The antiderivative is a function and answers a question about the whole curve; the definite integral is a number and answers one about a specific interval.
How to use this calculator
- Type the polynomial in x Exponents with ^, terms joined by plus and minus. Coefficients may be negative or fractional and terms may be in any order.
- Leave both limits blank for the antiderivative The answer then comes back as a polynomial with a constant of integration attached. This is the mode for an indefinite integral.
- Fill in both limits for a definite integral Both are needed. Supplying only one leaves the tool in indefinite mode rather than assuming a value for the other.
- Read the per-term lines Each term shows its own application of the reverse power rule, which is where a coefficient error localises.
The formula, and where it comes from
∫ c·xⁿ dx = c/(n+1)·xⁿ⁺¹ + C ∫ₐᵇ f = F(b) − F(a)
The reverse power rule inverts differentiation exactly. Raising the exponent undoes the reduction, and dividing by the new exponent undoes the multiplication — which is why the divisor is the raised power rather than the original one.
Checking the result costs nothing: differentiate it and the integrand must come back. Multiplying the new coefficient by the new exponent restores the old coefficient, and reducing the power restores the old power. That round trip is the definition of an antiderivative.
The constant of integration is not optional. Differentiation destroys constants, so every function has infinitely many antiderivatives differing by one, and the polynomial reported is a single representative of that family.
For a definite integral the antiderivative is found first and then evaluated at both limits, with the difference reported. The constant cancels between the two evaluations, which is why it never appears in a definite answer and why any antiderivative works equally well.
What each input means
- f(x) Integrand — form field “Polynomial in x”
- The polynomial to integrate, in x with non-negative whole exponents. It is parsed into a coefficient list before the rule is applied.
- a Lower limit — form field “Lower limit a (optional)”
- Optional. A finite number; leave it blank together with the upper limit for the indefinite integral.
- b Upper limit — form field “Upper limit b (optional)”
- Optional, and used only when the lower limit is also given. Swapping the two negates the result.
- F(x) Antiderivative
- One degree higher than the integrand. Differentiating it returns the integrand exactly.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
An indefinite integral
A quadratic integrand with three terms, and no limits supplied. The answer is a cubic plus the constant.
Inputs Polynomial in x = 6x^2 - 4x + 3, Lower limit a (optional) = , Upper limit b (optional) =
- Integrand f(x) = 6x² − 4x + 3
- Reverse power rule ∫ 3x dx = 3x¹
- Reverse power rule ∫ -4x dx = -2x²
- Reverse power rule ∫ 6x² dx = 2x³
- Indefinite integral ∫ f(x) dx = 2x³ − 2x² + 3x + C
Result ∫ f(x) dx = 2x³ − 2x² + 3x + C
The leading coefficient of 6 divided by the new exponent of 3 gives 2, so the awkward number disappears entirely. Integrands are often chosen this way in textbooks, which can build a false expectation that the division always tidies up.
The constant term integrates to a multiple of x rather than vanishing — the mirror image of differentiation, where it was the constant that disappeared. Every term gains a power here rather than losing one.
The same integrand with limits
The identical polynomial evaluated from 0 to 2. The antiderivative is found first and then evaluated at each end.
Inputs Polynomial in x = 6x^2 - 4x + 3, Lower limit a (optional) = 0, Upper limit b (optional) = 2
- Integrand f(x) = 6x² − 4x + 3
- Reverse power rule ∫ 3x dx = 3x¹
- Reverse power rule ∫ -4x dx = -2x²
- Reverse power rule ∫ 6x² dx = 2x³
- Indefinite integral ∫ f(x) dx = 2x³ − 2x² + 3x + C
- Evaluate F(b) F(2) = 14
- Evaluate F(a) F(0) = 0
- Definite integral F(b) − F(a) = 14
Result ∫ from 0 to 2 = 14
The antiderivative line is unchanged from the indefinite case, and two evaluation lines follow it before the subtraction. Seeing both values separately is what makes a sign error in the difference obvious.
The lower limit of zero makes its evaluation vanish here, which is a common convenience rather than a rule. With a non-zero lower limit both evaluations matter and the subtraction has to be done carefully.
Reading the result
The degree always rises by one
Integrating a polynomial of degree n gives one of degree n + 1. An answer of the same degree as the integrand means an exponent was never raised, which is the mirror of the corresponding error in differentiation.
A definite result is signed, not an area
Stretches where the polynomial is below the axis contribute negatively, so a result near zero can mean cancellation rather than a small region. Geometric area needs the interval split at each root.
Any antiderivative gives the same definite answer
Adding a constant shifts both evaluations equally and the difference is unaffected. That is why the constant is dropped from a definite calculation without any loss.
When you would use this
Area under a polynomial curve
Where a quantity is modelled by a polynomial, the definite integral over an interval gives the accumulated total exactly — no sampling error, unlike a numerical method.
Recovering a function from its rate of change
A polynomial rate integrates to the quantity itself, up to the constant. Supplying one known value pins that constant down and completes the reconstruction.
Assumptions and limitations
What this calculator assumes
- The integrand is a polynomial in x with non-negative whole exponents.
- Integration is with respect to x and the result is exact.
- A definite integral requires both limits as finite numbers; one alone is ignored.
- Terms are handled independently, which linearity of the integral guarantees is valid.
Where it stops being the right tool
- Polynomials only. Trigonometric, exponential and logarithmic integrands need the general symbolic integral calculator.
- No negative exponents, so the case that produces a logarithm never arises here.
- No substitution or integration by parts, since neither is ever required for a polynomial.
- Improper integrals are out of scope: both limits must be finite.
Common mistakes
Dividing by the original exponent
Why it happens. Both numbers are present and the rule reads as a single motion, so it is easy to divide by the power that was there rather than the one just created.
How to avoid it. Do it in two steps. Write the raised exponent first, then divide by exactly what you have written — never by what you started with.
Omitting the constant of integration
Why it happens. The polynomial looks complete without it, and it contributes nothing to a definite integral, so the habit of leaving it out survives into indefinite work where it matters.
How to avoid it. Write it on every indefinite answer. Without it the result names one antiderivative rather than the whole family, which matters as soon as an initial condition is applied.
Reading a signed integral as an area
Why it happens. Area under the curve is the usual phrase, so a small or negative result for a curve that visibly encloses a lot of space reads as an error.
How to avoid it. Find where the polynomial crosses the axis and integrate each stretch separately, then add the magnitudes.
Key terms
Frequently asked questions
How do I get a definite integral?
Fill in both the lower and upper limits. The antiderivative is found first, then evaluated at each limit, and the difference reported. Leaving either blank returns the indefinite integral instead.
Why is there a + C?
Because differentiation destroys constants, so infinitely many polynomials share the same derivative. The constant records that the true antiderivative may differ from the one printed by any fixed amount.
Is the result exact?
Yes, throughout. Polynomial integration is symbolic — one division per term — and the definite evaluation is the exact difference of two exact values, with no sampling or approximation anywhere.
How can I check the antiderivative?
Differentiate it. Multiplying each coefficient by its exponent and reducing the power must return the original integrand exactly, which is what being an antiderivative means.