Antiderivatives and Indefinite Integrals
- Calculus 3.8
Antiderivatives
A function $F$ is called an antiderivative of $f$ on an interval $I$ if $F’(x) = f(x)$ for all $x$ in $I$.
We can easily find that the antiderivative is not unique. From the corollary of the MVT, functions with same derivatives can differ by a constant. Thus, if $F(x)$ is an antiderivative of $f(x)$, $F(x)+C$ is also an antiderivative of $f(x)$ for an arbitrary constant $C$. Therefore, the most general antiderivative of $f$ on $I$ is a family of functions given by:
\[ F(x)+C \]
whose graphs are vertical translations of one another. Antiderivatives are usually denoted as the upper case letters of the function while the original function is usually denoted as the lower case letters.
Indefinite Integrals
The collection of all antiderivatives of a function $f$ is called the indefinite integral of $f$ respect to $x$, and is denoted by:
\[ \int f(x) \dd{x} \]
The symbol $\int$ is called the integral sign, and the function $f(x)$ is called the integrand. $x$ is the variable of integration and $\dd{x}$ is called the differential of $x$.
Properties
\[ \begin{align*} & \int kf(x) \dd{x} = k\int f(x) \dd{x} \nl & \int (f(x)+g(x)) \dd{x} = \int f(x) \dd{x} + \int g(x) \dd{x} \end{align*} \]
These linear properties simply follow from the linearity of the derivative operator.
Examples
| $f(x)$ | $x^n \;\; (-1\ne n \in \mathbb{Z})$ | $\sin x$ | $\cos x$ | $\sec^2 x$ | $\csc^2 x$ | $\sec x \tan x$ | $\csc x \cot x$ |
|---|---|---|---|---|---|---|---|
| $\dps \int f(x) \dd{x}$ | $\dfrac{x^{n+1}}{n+1} + C$ | $-\cos x + C$ | $\sin x + C$ | $\tan x + C$ | $-\cot x + C$ | $\sec x + C$ | $-\csc x + C$ |
