# A use exercise 105 to show that the th coefficient is

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Chapter 0 / Exercise 144
College Algebra
Gustafson/Hughes
Expert Verified
(a) Use Exercise 105 to show that the th coefficient given by (b) Let Find and a 3 . a 2 , a 1 , f x a n 1 f x sin nx dx a n n a 1 sin x a 2 sin 2 x a 3 sin 3 x . . . a N sin f x N i 1 a i sin ix sin x , sin 2 x , sin 3 x , . . . , cos x , cos 2 x , cos 3 x , , . f , g 0. g f f , g b a f x g x dx a , b g 2 0 cos n x dx 1 2 3 4 5 6 . . . n 1 n n v 2 , n 2 0 cos n x dx 2 3 4 5 6 7 . . . n 1 n v 3 , n L t H t b 1 1 6 12 0 f t sin t 6 dt a 1 1 6 12 0 f t cos t 6 dt a 0 1 12 12 0 f t dt b 1 a 1 , a 0 , t 0 f t a 0 a 1 cos t 6 b 1 sin t sin 4 x cos 2 x dx sec 4 2 x 5 dx cos 4 x dx sin 5 x dx Month Jan Feb Mar Apr May Jun Max 33.5 35.4 44.7 55.6 67.4 76.2 Min 20.3 20.9 28.2 37.9 48.7 58.5 Month Jul Aug Sep Oct Nov Dec Max 80.4 79.0 72.0 61.0 49.3 38.6 Min 63.7 62.7 55.9 45.5 36.4 26.8 is x . . Nx . . . . f 2 . n . 6
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Chapter 0 / Exercise 144
College Algebra
Gustafson/Hughes
Expert Verified
SECTION 8.4 Trigonometric Substitution 543 Section 8.4 Trigonometric Substitution Use trigonometric substitution to solve an integral. Use integrals to model and solve real-life applications. Trigonometric Substitution Now that you can evaluate integrals involving powers of trigonometric functions, you can use trigonometric substitution to evaluate integrals involving the radicals and The objective with trigonometric substitution is to eliminate the radical in the integrand. You do this with the Pythagorean identities For example, if let where Then Note that because NOTE The restrictions on ensure that the function that defines the substitution is one-to-one. In fact, these are the same intervals over which the arcsine, arctangent, and arcsecant are defined. 2 2. cos 0, a cos . a 2 cos 2 a 2 1 sin 2 a 2 u 2 a 2 a 2 sin 2 2 2. u a sin , a > 0, u 2 a 2 . a 2 u 2 , a 2 u 2 , and tan 2 sec 2 1. sec 2 1 tan 2 , cos 2 1 sin 2 , Trigonometric Substitution 1. For integrals involving let Then where 2. For integrals involving let Then where 3. For integrals involving let Then where or Use the positive value if and the negative value if u < a . u > a 2 < . 0 < 2 u 2 a 2 ± a tan , u a sec . a u θ u 2 - a 2 u 2 a 2 , 2 < < 2. a 2 u 2 a sec , u a tan . a u θ a 2 + u 2 a 2 u 2 , 2 2. a 2 u 2 a cos , u a sin . a u θ a 2 - u 2 a 2 u 2 , a > 0 E X P L O R A T I O N Integrating a Radical Function Up to this point in the text, you have not evaluated the following integral. From geometry, you should be able to find the exact value of this integral— what is it? Using numerical integra- tion, with Simpson’s Rule or the Trapezoidal Rule, you can’t be sure of the accuracy of the approximation. Why? Try finding the exact value using the substitution and Does your answer agree with the value you obtained using geometry? dx cos d . x sin 1 1 1 x 2 dx
. 544 CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals EXAMPLE 1 Trigonometric Substitution: Find Solution First, note that none of the basic integration rules applies. To use trigono- metric substitution, you should observe that is of the form So, you can use the substitution Using differentiation and the triangle shown in Figure 8.6, you obtain and So, trigonometric substitution yields Substitute.