📂Calculus

Antiderivatives and Indefinite Integrals

Definition

A function $F$ is called an antiderivative of another function $f$ if it satisfies $F^{\prime} = f$.

Explanation

An antiderivative is translated as 원시함수 (primitive function), 역도함수 (reverse derivative), etc.

The process of finding a function $F$ that satisfies $F^{\prime} = f$ for a given $f$, or the function $F$ itself, is called an indefinite integral. The indefinite integral or antiderivative of $f$ is expressed as follows:

$$\int f(x)dx$$

Since a constant differentiated becomes $0$, if a function $F$ is an indefinite integral of $f$, then $F + C$ is also an indefinite integral of $f$ for any arbitrary constant $C$. In other words, there are infinitely many indefinite integrals for a given function. Thus, when expressing an indefinite integral, it is written as follows:

$$\int f(x)dx = F(x) + C$$