Chapter 4 | Applications of Derivatives
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4.50 Verify that ∫ x cos xdx = x sin x +cos x + C .
In Table 4.13 , 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 Example 4.51 a. we showed that an antiderivative of the sum x + e x is given by the sum ⎛ ⎝ x 2 2 ⎞ ⎠ + 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 d dx ( F ( x )+ G ( x )) = F ′( x )+ G ′( x ) = f ( x )+ g ( x ). Therefore, F ( x )+ G ( x ) is an antiderivative of f ( x )+ g ( x ) and we have ∫ ⎛ ⎝ f ( x )+ g ( x ) ⎞ ⎠ dx = F ( x )+ G ( x )+ C . Similarly, ∫ ⎛ ⎝ f ( x )− g ( x ) ⎞ ⎠ dx = F ( x )− G ( x )+ C . In addition, consider the task of finding an antiderivative of kf ( x ), where k is any real number. Since d dx ⎛ ⎝ kf ( x ) ⎞ ⎠ = k d dx F ( x ) = kf ′( x ) for any real number k , we conclude that ∫ kf ( x ) dx = kF ( x )+ C . These properties are summarized next.
Theorem 4.16: Properties of Indefinite Integrals Let F and G be antiderivatives of f and g , respectively, and let k be any real number. Sums and Differences ∫ ⎛ ⎝ f ( x )± g ( x ) ⎞ ⎠ dx = F ( x )± G ( x )+ C Constant Multiples ∫ kf ( x ) dx = kF ( x )+ C
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 Example 4.51 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. Example 4.52 Evaluating Indefinite Integrals
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