Polynomial Integral Calculator
The Polynomial Integral Calculator computes the indefinite integral (antiderivative) of any polynomial in seconds using the Power Rule. Given a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, its indefinite integral is: ∫P(x)dx = aₙxⁿ⁺¹/(n+1) + aₙ₋₁xⁿ/n + … + a₁x²/2 + a₀x + C Provide the coefficients from highest to lowest degree (e.g., "3, 0, -2, 1" for 3x³−2x+1) and the tool returns the complete antiderivative expression. Used daily in calculus courses, physics (finding position from velocity), and engineering (computing areas under polynomial curves).
To integrate a polynomial, apply the Power Rule to each term: ∫axⁿ dx = a·xⁿ⁺¹/(n+1) + C. For example, ∫(3x² + 2x + 5) dx = x³ + x² + 5x + C. Always append +C (the constant of integration). Enter the coefficients from highest to lowest degree, separated by commas — e.g., "3, 2, 5" for 3x²+2x+5.
When to use this calculator
- Finding the position function s(t) from a velocity polynomial v(t) in kinematics — e.g., integrating v(t)=6t²−4t+1 to get s(t)=2t³−2t²+t+C.
- Computing the exact area under a polynomial curve between two bounds using the Fundamental Theorem of Calculus after finding the antiderivative.
- Solving initial-value problems in differential equations where the right-hand side is a polynomial, such as y'=4x³−3x+7, requiring y=x⁴−(3/2)x²+7x+C.
- Determining total revenue or total cost functions in economics when the marginal cost/revenue function is expressed as a polynomial of output quantity q.
Worked Example
- Polynomial: 3x² + 2x + 5 → Coefficients: 3, 2, 5
- 3x² → 3x³/3 = x³
- 2x → 2x²/2 = x²
- 5 → 5x¹/1 = 5x
- ∫(3x²+2x+5)dx = x³ + x² + 5x + C
How it works
2 min readHow It's Calculated
The Power Rule for Integration is the core formula. For every term aₙxⁿ (where n ≠ −1):
∫aₙxⁿ dx = aₙ · xⁿ⁺¹ / (n+1) + CFor a full polynomial with coefficients entered from highest to lowest degree:
Input: [c_n, c_{n-1}, ..., c_1, c_0] (length = n+1 terms)
For each index i (0-based from left), the degree of that term is:
degree(i) = n - i where n = len(coefficients) - 1
Integrated coefficient:
new_coef(i) = c_i / (degree(i) + 1)
New degree:
new_degree(i) = degree(i) + 1
Result: sum of all new_coef(i) · x^new_degree(i) + C---
Power Rule Reference Table
The table below shows the integral of the most common polynomial terms — bookmark it for quick reference in exams and homework.
| Term f(x) | Degree | ∫f(x) dx | Result |
|---|---|---|---|
| 7 | 0 | 7x + C | constant rule |
| 3x | 1 | (3/2)x² + C | |
| x² | 2 | x³/3 + C | |
| −4x³ | 3 | −x⁴ + C | |
| 5x⁴ | 4 | x⁵ + C | |
| 2x⁵ | 5 | (1/3)x⁶ + C | |
| −x⁶ | 6 | −x⁷/7 + C | |
| ax^n | n | a·xⁿ⁺¹/(n+1) + C | general rule; n ≠ −1 |
| 3x²+2x+5 | 2 | x³+x²+5x + C | full quadratic |
| 4x³−3x+1 | 3 | x⁴−(3/2)x²+x + C | cubic, zero middle |
| 6t²−10t+2 | 2 | 2t³−5t²+2t + C | kinematics: v→s |
> Note: The rule n ≠ −1 matters! For x⁻¹ = 1/x, the integral is ln|x|+C — not handled by this polynomial calculator.
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Typical Examples
Example 1 — Quadratic: 3x²+2x+5
Example 2 — Cubic with zero middle term: 4x³+0x²−3x+1
Example 3 — Kinematics (velocity → position)
Example 4 — Constant function
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Common Errors
1. Forgetting +C — The indefinite integral is a family of functions. Omitting C is marked wrong on every calculus exam.
2. Wrong coefficient order — This calculator expects highest to lowest degree. Entering "1,2" gives x²+2x+C (polynomial x+2), not 2x²+x+C (polynomial 2x+1).
3. Skipping zero coefficients — For x⁴−9, you must enter 1, 0, 0, 0, −9, not "1, −9".
4. Dividing by the old exponent — For 6x², the integral is 6x³/3 = 2x³, NOT 6x²/2 = 3x².
5. Using for non-polynomial terms — 1/x (= x⁻¹) requires the logarithm rule; √x = x^(1/2) still follows the power rule but is not a standard polynomial.
Frequently asked questions
What is the Power Rule for Integration?
The Power Rule states ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for any n ≠ −1. It works because differentiation and integration are inverse operations: d/dx[xⁿ⁺¹/(n+1)] = xⁿ, confirming the antiderivative. This is a direct consequence of the Fundamental Theorem of Calculus.
Why do I always add '+C' to an indefinite integral?
Because the derivative of any constant is 0, infinitely many functions share the same derivative. For example, d/dx[x²+3] = d/dx[x²−100] = 2x. Without +C, you're claiming one specific antiderivative rather than the entire family. In applied problems, C is pinned down by an initial condition (e.g., position at t=0).
How do I enter a polynomial that has missing (zero) terms?
Include a 0 for every missing degree. For x⁴−9, which has no x³, x², or x terms, enter '1, 0, 0, 0, −9'. Skipping zeros shifts all degree assignments and produces a completely wrong integral. Rule: number of coefficients = highest degree + 1.
What is the integral of a constant, like ∫7 dx?
A constant 'a' is treated as a·x⁰, so the power rule gives a·x¹/1 = ax. Thus ∫7 dx = 7x+C. Geometrically: the area under y=7 from 0 to b is a rectangle with area 7b.
How do I use the result to compute a definite integral?
Once you have the antiderivative F(x), apply the Fundamental Theorem of Calculus: ∫a to b] P(x)dx = F(b) − F(a). For example, ∫[0 to 2dx = [x³+x²+5x]₀² = (8+4+10) − 0 = 22. The +C cancels in the subtraction.
How can I verify my computed antiderivative?
Differentiate the result. If F(x) is your antiderivative, then F'(x) must equal P(x) exactly. For instance, if ∫(2x+1)dx = x²+x+C, then d/dx[x²+x+C] = 2x+1 ✓. This differentiation check catches every arithmetic mistake.
Can this calculator handle fractional or negative exponents?
No — the tool is designed for polynomials with non-negative integer exponents. The power rule still applies to x^(1/2) → (2/3)x^(3/2)+C, but that is not a standard polynomial. For x⁻¹ = 1/x, the integral is ln|x|+C, which requires a completely different rule.
How does polynomial integration appear in real-world physics?
In kinematics, if acceleration a(t) is a polynomial, integrating gives velocity v(t)=∫a(t)dt, and integrating again gives position s(t)=∫v(t)dt. Constant acceleration a=9.8 m/s² integrates to v(t)=9.8t+v₀ and then to s(t)=4.9t²+v₀t+s₀ — the classic projectile equation.
What is the difference between indefinite and definite integrals?
An indefinite integral returns a family of functions (e.g., x²+x+C), while a definite integral computes a specific numeric value — the net signed area between the curve and the x-axis on an interval [a, b] — using F(b)−F(a).