Integral Calculator
Find the indefinite integral of a polynomial with the reverse power rule.
Reverse power rule, plus the constant of integration.
Integration reverses the power rule: ∫a·xⁿ dx = a ÷ (n+1) · xⁿ⁺¹ + C — raise the exponent by one and divide by it. So ∫(6x − 5) dx = 3x² − 5x + C, whose derivative is 6x − 5. The “+ C” is required for indefinite integrals.
The reverse power rule
Integration undoes differentiation. Where the power rule lowers an exponent, the reverse power rule raises it: add one to the exponent, then divide by that new exponent. Because many functions share the same derivative, an indefinite integral always ends with a constant of integration, “+ C”.
raise the exponent by one, divide by the new exponent; valid for n ≠ −1
Worked example
Integrate ∫ (6x − 5) dx:
- 1 Integrate 6x (here n = 1). Raise to x²; divide by the new exponent 2: 6 ÷ 2 · x² = 3x².
- 2 Integrate −5 (here n = 0). Raise to x¹; divide by 1: −5 ÷ 1 · x = −5x.
- 3 Add the constant. Result: 3x² − 5x + C. Differentiating it gives back 6x − 5. ✓
Common integral rules
The standard antiderivatives; every indefinite integral adds + C.
| Function f(x) | Integral ∫ f(x) dx |
|---|---|
| xⁿ (n ≠ −1) | xⁿ⁺¹ ÷ (n + 1) + C |
| 1 ÷ x | ln|x| + C |
| eˣ | eˣ + C |
| cos x | sin x + C |
| sin x | −cos x + C |
| constant k | k·x + C |
The “+ C” and the n = −1 exception
Why “+ C”? Any two antiderivatives of the same function differ only by a constant — the derivative of a constant is zero — so an indefinite integral represents a whole family of curves.
The rule breaks at n = −1. You can’t divide by n + 1 = 0. The integral of 1 ÷ x is instead ln|x| + C, a special case worth memorising.
Definite integrals. To find an area, evaluate the antiderivative at the upper and lower limits and subtract; the constant C cancels out. This tool returns the indefinite integral.