Polynomial Integral Calculator

Calculator inputs

Coefficients, highest degree first

Example input 3, -2, 5, -7 means 3x³ - 2x² + 5x - 7.

Variable symbol

Decimal precision

Lower bound

Upper bound

Evaluate F(x) and P(x) at

Sample rows in result table

Example data table

This example uses the polynomial 3x³ - 2x² + 5x - 7 and antiderivative 0.75x⁴ - 0.666667x³ + 2.5x² - 7x + C.

x	P(x)	F(x) without C
-2	-45	38.666667
0	-7	0
2	19	-1.333333
4	157	106.666667

Formula used

If P(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀, then the integral is computed term by term.

General rule:

\int a k x k dx = (a k / (k + 1))x k+1 + C

Polynomial antiderivative:

\intP(x)dx = \sum (a k / (k + 1))x k+1 + C

Definite integral on [a, b]:

\int a b P(x)dx = F(b) - F(a)

How to use this calculator

Enter coefficients from highest degree to constant term.
Set the variable symbol and desired decimal precision.
Add lower and upper bounds for a definite integral.
Add an evaluation point to inspect P(x) and F(x).
Press Integrate polynomial to display the result above the form.
Review the antiderivative, interval values, table, and graph.
Use the export buttons to save CSV and PDF copies.

FAQs

1. What coefficient order should I enter?

Enter coefficients from the highest degree term down to the constant term. For 4x² - 3x + 9, use 4, -3, 9.

2. Can this calculator compute indefinite and definite integrals?

Yes. It always computes the antiderivative. It also computes a definite integral when both lower and upper bounds are supplied.

3. Why does the result include C?

The constant C represents the family of antiderivatives. Every indefinite integral of the same polynomial differs only by a constant.

4. What happens if I reverse the bounds?

The calculator reorders them so the lower bound comes first. A notice appears in the result section when this automatic adjustment occurs.

5. What does the evaluation point do?

It calculates the original polynomial value P(x) and the antiderivative value F(x) at a chosen x-value. This helps with checks and interpretation.

6. Does the graph show both functions?

Yes. The Plotly graph plots the original polynomial and the antiderivative together, helping you compare slope behavior and accumulated change.

7. What is the average value on the interval?

It is the definite integral divided by interval length. This gives the mean function value across the selected interval.

8. Can I export the results?

Yes. CSV export saves summary metrics and sampled points. PDF export captures the visible result block, including the graph and table.