# 4.10 Antiderivatives  (Page 2/10)

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We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.

## Finding antiderivatives

For each of the following functions, find all antiderivatives.

1. $f\left(x\right)=3{x}^{2}$
2. $f\left(x\right)=\frac{1}{x}$
3. $f\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x$
4. $f\left(x\right)={e}^{x}$
1. Because
$\frac{d}{dx}\left({x}^{3}\right)=3{x}^{2}$

then $F\left(x\right)={x}^{3}$ is an antiderivative of $3{x}^{2}.$ Therefore, every antiderivative of $3{x}^{2}$ is of the form ${x}^{3}+C$ for some constant $C,$ and every function of the form ${x}^{3}+C$ is an antiderivative of $3{x}^{2}.$
2. Let $f\left(x\right)=\text{ln}|x|.$ For $x>0,f\left(x\right)=\text{ln}\left(x\right)$ and
$\frac{d}{dx}\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x\right)=\frac{1}{x}.$

For $x<0,f\left(x\right)=\text{ln}\left(\text{−}x\right)$ and
$\frac{d}{dx}\left(\text{ln}\left(\text{−}x\right)\right)=-\frac{1}{\text{−}x}=\frac{1}{x}.$

Therefore,
$\frac{d}{dx}\left(\text{ln}|x|\right)=\frac{1}{x}.$

Thus, $F\left(x\right)=\text{ln}|x|$ is an antiderivative of $\frac{1}{x}.$ Therefore, every antiderivative of $\frac{1}{x}$ is of the form $\text{ln}|x|+C$ for some constant $C$ and every function of the form $\text{ln}|x|+C$ is an antiderivative of $\frac{1}{x}.$
3. We have
$\frac{d}{dx}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x,$

so $F\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ is an antiderivative of $\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ Therefore, every antiderivative of $\text{cos}\phantom{\rule{0.1em}{0ex}}x$ is of the form $\text{sin}\phantom{\rule{0.1em}{0ex}}x+C$ for some constant $C$ and every function of the form $\text{sin}\phantom{\rule{0.1em}{0ex}}x+C$ is an antiderivative of $\text{cos}\phantom{\rule{0.1em}{0ex}}x.$
4. Since
$\frac{d}{dx}\left({e}^{x}\right)={e}^{x},$

then $F\left(x\right)={e}^{x}$ is an antiderivative of ${e}^{x}.$ Therefore, every antiderivative of ${e}^{x}$ is of the form ${e}^{x}+C$ for some constant $C$ and every function of the form ${e}^{x}+C$ is an antiderivative of ${e}^{x}.$

Find all antiderivatives of $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x.$

$\text{−}\text{cos}\phantom{\rule{0.1em}{0ex}}x+C$

## Indefinite integrals

We now look at the formal notation used to represent antiderivatives and examine some of their properties. These properties allow us to find antiderivatives of more complicated functions. Given a function $f,$ we use the notation ${f}^{\prime }\left(x\right)$ or $\frac{df}{dx}$ to denote the derivative of $f.$ Here we introduce notation for antiderivatives. If $F$ is an antiderivative of $f,$ we say that $F\left(x\right)+C$ is the most general antiderivative of $f$ and write

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

The symbol $\int$ is called an integral sign , and $\int f\left(x\right)dx$ is called the indefinite integral of $f.$

## Definition

Given a function $f,$ the indefinite integral    of $f,$ denoted

$\int f\left(x\right)dx,$

is the most general antiderivative of $f.$ If $F$ is an antiderivative of $f,$ then

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

The expression $f\left(x\right)$ is called the integrand and the variable $x$ is the variable of integration .

Given the terminology introduced in this definition, the act of finding the antiderivatives of a function $f$ is usually referred to as integrating $f.$

For a function $f$ and an antiderivative $F,$ the functions $F\left(x\right)+C,$ where $C$ is any real number, is often referred to as the family of antiderivatives of $f.$ For example, since ${x}^{2}$ is an antiderivative of $2x$ and any antiderivative of $2x$ is of the form ${x}^{2}+C,$ we write

$\int 2x\phantom{\rule{0.1em}{0ex}}dx={x}^{2}+C.$

The collection of all functions of the form ${x}^{2}+C,$ where $C$ is any real number, is known as the family of antiderivatives of $2x.$ [link] shows a graph of this family of antiderivatives. The family of antiderivatives of 2 x consists of all functions of the form x 2 + C , where C is any real number.

For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for $n\ne \text{−}1,$

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C,$

which comes directly from

$\frac{d}{dx}\left(\frac{{x}^{n+1}}{n+1}\right)=\left(n+1\right)\frac{{x}^{n}}{n+1}={x}^{n}.$

This fact is known as the power rule for integrals .

## Power rule for integrals

For $n\ne \text{−}1,$

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C.$

Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in Appendix B .

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