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Differentiation Formula | Indefinite Integral |
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$\frac{d}{dx}\left(k\right)=0$ | $\int kdx={\displaystyle \int k{x}^{0}dx}}=kx+C$ |
$\frac{d}{dx}\left({x}^{n}\right)=n{x}^{n-1}$ | $\int {x}^{n}dn}=\frac{{x}^{n+1}}{n+1}+C$ for $n\ne \text{\u2212}1$ |
$\frac{d}{dx}\left(\text{ln}\left|x\right|\right)=\frac{1}{x}$ | $\int \frac{1}{x}dx=\text{ln}\left|x\right|}+C$ |
$\frac{d}{dx}\left({e}^{x}\right)={e}^{x}$ | $\int {e}^{x}dx}={e}^{x}+C$ |
$\frac{d}{dx}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x$ | $\int \text{cos}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx=\text{sin}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212}\text{sin}\phantom{\rule{0.1em}{0ex}}x$ | $\int \text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx=\text{\u2212}\text{cos}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left(\text{tan}\phantom{\rule{0.1em}{0ex}}x\right)={\text{sec}}^{2}x$ | $\int {\text{sec}}^{2}x\phantom{\rule{0.1em}{0ex}}dx=\text{tan}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left(\text{csc}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212}\text{csc}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}x$ | $\int \text{csc}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx=\text{\u2212}\text{csc}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left(\text{sec}\phantom{\rule{0.1em}{0ex}}x\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}x$ | $\int \text{sec}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx=\text{sec}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left(\text{cot}\phantom{\rule{0.1em}{0ex}}x\right)=\text{\u2212}{\text{csc}}^{2}x$ | $\int {\text{csc}}^{2}x\phantom{\rule{0.1em}{0ex}}dx=\text{\u2212}\text{cot}\phantom{\rule{0.1em}{0ex}}x+C$ |
$\frac{d}{dx}\left({\text{sin}}^{\mathrm{-1}}x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$ | $\int \frac{1}{\sqrt{1-{x}^{2}}}={\text{sin}}^{\mathrm{-1}}x+C$ |
$\frac{d}{dx}\left({\text{tan}}^{\mathrm{-1}}x\right)=\frac{1}{1+{x}^{2}}$ | $\int \frac{1}{1+{x}^{2}}dx={\text{tan}}^{\mathrm{-1}}x+C$ |
$\frac{d}{dx}\left({\text{sec}}^{\mathrm{-1}}\left|x\right|\right)=\frac{1}{x\sqrt{{x}^{2}-1}}$ | $\int \frac{1}{x\sqrt{{x}^{2}-1}}dx={\text{sec}}^{\mathrm{-1}}\left|x\right|+C$ |
From the definition of indefinite integral of $f,$ we know
if and only if $F$ is an antiderivative of $f.$ Therefore, when claiming that
it is important to check whether this statement is correct by verifying that ${F}^{\prime}\left(x\right)=f\left(x\right).$
Each of the following statements is of the form $\int f\left(x\right)dx=F\left(x\right)+C}.$ Verify that each statement is correct by showing that ${F}^{\prime}\left(x\right)=f\left(x\right).$
Verify that $\int x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}dx=x\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\phantom{\rule{0.1em}{0ex}}x+C.$
$\frac{d}{dx}\left(x\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\phantom{\rule{0.1em}{0ex}}x+C\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x+x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x-\text{sin}\phantom{\rule{0.1em}{0ex}}x=x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$
In [link] , we listed the indefinite integrals for many elementary functions. Let’s now turn our attention to evaluating indefinite integrals for more complicated functions. For example, consider finding an antiderivative of a sum $f+g.$ In [link] a. we showed that an antiderivative of the sum $x+{e}^{x}$ is given by the sum $\left(\frac{{x}^{2}}{2}\right)+{e}^{x}$ —that is, an antiderivative of a sum is given by a sum of antiderivatives. This result was not specific to this example. In general, if $F$ and $G$ are antiderivatives of any functions $f$ and $g,$ respectively, then
Therefore, $F\left(x\right)+G\left(x\right)$ is an antiderivative of $f\left(x\right)+g\left(x\right)$ and we have
Similarly,
In addition, consider the task of finding an antiderivative of $kf\left(x\right),$ where $k$ is any real number. Since
for any real number $k,$ we conclude that
These properties are summarized next.
Let $F$ and $G$ be antiderivatives of $f$ and $g,$ respectively, and let $k$ be any real number.
Sums and Differences
Constant Multiples
From this theorem, we can evaluate any integral involving a sum, difference, or constant multiple of functions with antiderivatives that are known. Evaluating integrals involving products, quotients, or compositions is more complicated (see [link] b. for an example involving an antiderivative of a product.) We look at and address integrals involving these more complicated functions in Introduction to Integration . In the next example, we examine how to use this theorem to calculate the indefinite integrals of several functions.
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