Polynomial Definite Integral Calculator
Integrate a polynomial of degree up to six between two bounds. See an exact fraction answer, the antiderivative and endpoint substitution, with a shaded curve.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Increase each power by one and divide its coefficient by that new power to obtain an antiderivative. Evaluate it at the upper and lower bounds, then subtract: F(upper)−F(lower). The additive constant cancels.
∫axⁿ dx=axⁿ⁺¹/(n+1)+C; ∫ₐᵇf(x)dx=F(b)−F(a)
Worked example
For f(x)=3x², an antiderivative is F(x)=x³. Between 0 and 2, the definite integral is 2³−0³=8. By contrast, ∫ from −1 to 1 of x dx=0 because the positive and negative signed contributions cancel; the total geometric area is not zero.
Assumptions and input details
Enter 1–7 coefficients in descending powers, including zeros for missing terms. Whole numbers, decimals with up to six places and fractions are accepted, using the same coefficient limits as the derivative tool. Bounds must be within ±1000 with reduced denominators at most 1e6. Reversed bounds negate the integral and equal bounds give zero. The result is a signed integral, not total absolute area; shading is a sampled illustration, not a numerical quadrature calculation. Only explicit polynomials of degree at most six are supported.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Polynomial antiderivatives and endpoint substitutions use exact fraction arithmetic for the accepted inputs. Decimal answers and curve shading are approximations.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Differentiate a polynomial of degree up to six, evaluate its slope at a point and find the tangent line. Enter coefficients and follow each power-rule term.
Find the maximum and minimum of ax²+bx+c on an entered closed interval. Compare both endpoints and the vertex, with a graph and candidate table.
Add, subtract, multiply or divide fractions and mixed numbers. Follow exact fraction steps, simplification and a number-line diagram.
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