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Chapter 08 - 5E-08(pp 510-519 5:19 PM Page 510 CHAPTER 8...

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Techniques of Integration The techniques of this chapter enable us to find the height of a rocket a minute after liftoff and to compute the escape velocity of the rocket. C H A P T E R 8 5E-08(pp 510-519) 1/17/06 5:19 PM Page 510
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Because of the Fundamental Theorem of Calculus, we can integrate a function if we know an antiderivative, that is, an indefinite integral. We summarize here the most impor- tant integrals that we have learned so far. In this chapter we develop techniques for using these basic integration formulas to obtain indefinite integrals of more complicated functions. We learned the most important method of integration, the Substitution Rule, in Section 5.5. The other gen- eral technique, integration by parts, is presented in Section 8.1. Then we learn methods that are special to particular classes of functions such as trigonometric func- tions and rational functions. Integration is not as straightforward as differentiation; there are no rules that absolutely guarantee obtaining an indefinite integral of a function. Therefore, in Section 8.5 we discuss a strategy for integration. |||| 8.1 Integration by Parts Every differentiation rule has a corresponding integration rule. For instance, the Substi- tution Rule for integration corresponds to the Chain Rule for differentiation. The rule that corresponds to the Product Rule for differentiation is called the rule for integration by parts. The Product Rule states that if and are differentiable functions, then d dx f x t x f x t x t x f x t f y 1 s a 2 x 2 dx sin 1 x a C y 1 x 2 a 2 dx 1 a tan 1 x a C y cot x dx ln sin x C y tan x dx ln sec x C y cosh x dx sinh x C y sinh x dx cosh x C y csc x cot x dx csc x C y sec x tan x dx sec x C y csc 2 x dx cot x C y sec 2 x dx tan x C y cos x dx sin x C y sin x dx cos x C y a x dx a x ln a C y e x dx e x C y 1 x dx ln x C n 1 y x n dx x n 1 n 1 C 511 5E-08(pp 510-519) 1/17/06 5:19 PM Page 511
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512 ❙❙❙❙ CHAPTER 8 TECHNIQUES OF INTEGRATION In the notation for indefinite integrals this equation becomes or We can rearrange this equation as Formula 1 is called the formula for integration by parts . It is perhaps easier to remem- ber in the following notation. Let and . Then the differentials are and , so, by the Substitution Rule, the formula for integration by parts becomes EXAMPLE 1 Find . SOLUTION USING FORMULA 1 Suppose we choose and . Then and . (For we can choose any antiderivative of .) Thus, using Formula 1, we have It’s wise to check the answer by differentiating it. If we do so, we get , as expected. SOLUTION USING FORMULA 2 Let Then and so x cos x sin x C x cos x y cos x dx y x sin x dx y x sin x dx x cos x y cos x dx v cos x du dx d v sin x dx u x x sin x x cos x sin x C x cos x y cos x dx x cos x y cos x dx y x sin x dx f x t x y t x f x dx t t t x cos x f x 1 t x sin x f x x y x sin x dx y u d v u v y v du 2 d v t x dx du f x dx v t x u f x y f x t x dx f x t x y t x f x dx 1 y f x t x dx y t x f x dx f x t x y f x t x t x f x dx f x t x u d√ u du |||| It is helpful to use the pattern: v du d v u 5E-08(pp 510-519) 1/17/06
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