Antidifferentiation

Antidifferentiation is the process of finding an antiderivative of a function. It is the inverse operation of differentiation: for a given function f, an antiderivative F satisfies F'(x) = f(x) on an interval. The result is not unique; all antiderivatives differ by an additive constant.

The indefinite integral ∫ f(x) dx denotes the family of all antiderivatives. The constant of integration C is included because differentiating a constant yields zero. The fundamental theorem of calculus links antidifferentiation and definite integration: if F is an antiderivative of f on [a, b], then ∫_a^b f(x) dx = F(b) − F(a).

Common methods for finding antiderivatives include applying basic rules such as ∫ x^n dx = x^{n+1}/(n+1) + C for n ≠ −1; ∫ e^{ax} dx = (1/a)e^{ax} + C. More advanced techniques include substitution (u-substitution), integration by parts, partial fractions, and trigonometric substitutions. Some integrals do not have elementary antiderivatives and are expressed in terms of special functions.

Examples: ∫ 3x^2 dx = x^3 + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C.

Applications of antidifferentiation include computing areas under curves, solving problems in physics and engineering (such as work and displacement), and deriving cumulative distributions in probability.