4.10 Antiderivatives  (Page 3/10)

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 $F^{\prime }\left(x\right)=f\left(x\right).$

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.

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.