Integral formulas
Reference of basic integrals (antiderivatives) with a live definite-integral calculator.
| function f(x) | integral ∫ f(x) dx |
|---|---|
| k (constant) | k·x + C |
| xⁿ (n ≠ −1) | xⁿ⁺¹/(n+1) + C exponent increases by one, then divide by it |
| 1/x | ln|x| + C exception to the power rule |
| eˣ | eˣ + C |
| aˣ | aˣ/ln a + C |
| sin x | −cos x + C |
| cos x | sin x + C |
| 1/cos²x | tan x + C |
| 1/√(1−x²) | arcsin x + C |
| 1/(1+x²) | arctan x + C |
Indefinite and definite integral
The formulas in the table give the indefinite integral (antiderivative) – hence the constant + C (the derivative of a constant is zero, so there are infinitely many antiderivatives). The definite integral then plugs in the bounds: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is the antiderivative. The result of a definite integral is a number – the area under the curve.
The integral as the reverse of the derivative
Integration is the reverse operation of differentiation: we look for a function whose derivative is the given function. So every integral formula can be verified backwards – differentiating the result gives back the original function. See it visually as an area on the definite integral and Riemann sums pages.
The calculation above uses a numerical method (Simpson's rule), so it works even for functions without a simple antiderivative formula.