Integral Calculator
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What Your Result Means
- ∫f(x)dx: The antiderivative (indefinite integral) of your polynomial. This is a new function whose derivative equals your original function f(x).
- + C (constant of integration): Because any constant vanishes when differentiated, the antiderivative includes an arbitrary constant C. A specific value for C requires an initial condition or boundary value.
- Steps: Each term is integrated independently using the power rule — the coefficient is divided by (exponent + 1) and the exponent is raised by one.
How This Calculator Works
You enter a polynomial in x using caret notation for exponents (e.g., 3x^2 + 2x + 1). The tool parses each term into a coefficient and power, applies the power rule — ∫axⁿ dx = (a/(n+1))x^(n+1) — to each term independently, and sums the results with a constant of integration. It supports polynomials only; trigonometric, exponential, and logarithmic functions are not handled.
Quick Questions
What is the power rule for integration?
For any term axⁿ where n ≠ −1, the integral is (a/(n+1))x^(n+1) + C. You increase the exponent by one and divide the coefficient by the new exponent. This reverses the power rule for differentiation.
Why does this only work for polynomials?
The power rule applies to terms of the form axⁿ. Functions like sin(x), eˣ, or ln(x) require different integration techniques (substitution, integration by parts, etc.) that this tool does not implement.
What does the + C mean?
C is the constant of integration. Since the derivative of any constant is zero, there are infinitely many antiderivatives for a given function, all differing by a constant. You need an initial condition to find the specific value of C.
Can I enter negative or fractional exponents?
The parser supports integer exponents entered with the caret symbol (e.g., x^3, x^-2). Fractional exponents are not currently supported — enter only whole-number powers.
Sources
- Wikipedia — Power Rule (Integration) (derivation and proof of the power rule for antiderivatives)
- Wikipedia — Antiderivative (definition, existence, and properties of indefinite integrals)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.