Antiderivative Calculator
Find polynomial antiderivatives with evaluation and definite integral options.
Model your function as f(x) = a·x⁴ + b·x³ + c·x² + d·x + e. Leave a coefficient at 0 if the term is missing.
How to Use
- Enter the coefficients of your polynomial (use 0 for missing powers).
- Optionally add an evaluation point or bounds for a definite integral.
- Click Calculate to integrate term-by-term.
- Review the antiderivative, optional evaluations, and the integration steps.
Integrating Polynomials
Polynomials are among the easiest functions to integrate because each term follows a simple power rule. Integrate each term independently and sum the results.
Power rule: ∫ a·xⁿ dx = a·xⁿ⁺¹ / (n + 1) for n ≠ -1.
Why +C Matters
Antiderivatives are defined up to an arbitrary constant because differentiation removes constant terms. Always append +C when reporting an indefinite integral.
If you have an initial condition such as F(x₀) = y₀ you can solve for C explicitly.
From Antiderivative to Definite Integral
Once you know F(x), the definite integral from a to b becomes F(b) − F(a). The calculator performs this subtraction automatically when both bounds are provided.
- Use bounds to compute the net area under the curve.
- Order matters: integrate from the lower bound to the upper bound.
- Swapping bounds multiplies the result by −1.
Frequently Asked Questions
- Can this calculator integrate higher-degree polynomials?
- It supports terms up to x⁴ to keep the interface simple. For higher degrees, you can split the polynomial or use a CAS tool.
- What if my function has fractions or decimals?
- Enter fractional or decimal coefficients directly. The calculator keeps six decimal places of precision in the results.
- How do I handle missing terms?
- Set the coefficient to 0 for any missing term. For example, x² + 5 becomes a = 0, b = 0, c = 1, d = 0, e = 5.