aqu o sanb Using Formula i and solving for the required integral, we get j sec's de - {(see x tan x + ln jsec r + tan x) + C Integrals such as the one in the preceding exanaple may seem very special but they occur froquently in applications of integration, as we will see in Chapter 8. Integrals of the form cor"x cse's de can be found by similar methods because of the identity I+ cot's = cse'x. Finally, we can make use of another set of trigonometric identities 2 To evaluate the integrals (a) i sin mx cos Er dr, (b) í sin mx sin nx dx, or (c) f cos mx cos Re da, use the coresponding identity: product identities are discussed pendis D. (a) sin A cos B= {sin(A - ) + sinta + B)] (b) sin A sin B - (cos(A - B) - coslA + B)) (c) cos A cos B-(cos(A - B) + cos(A + B)] EXAMPLE 9 Evaluate sin 4r cos Sr de. SOLUTION This integral could be evaluated using integration by parts, but it's easier to use the identity in Equation 2(a) as follows: S sin dx con Sr dr - lsin-x) + sin 9.] da =!j(-sin x + sin 94) dx
aqu o sanb Using Formula i and solving for the required integral, we get j sec's de - {(see x tan x + ln jsec r + tan x) + C Integrals such as the one in the preceding exanaple may seem very special but they occur froquently in applications of integration, as we will see in Chapter 8. Integrals of the form cor"x cse's de can be found by similar methods because of the identity I+ cot's = cse'x. Finally, we can make use of another set of trigonometric identities 2 To evaluate the integrals (a) i sin mx cos Er dr, (b) í sin mx sin nx dx, or (c) f cos mx cos Re da, use the coresponding identity: product identities are discussed pendis D. (a) sin A cos B= {sin(A - ) + sinta + B)] (b) sin A sin B - (cos(A - B) - coslA + B)) (c) cos A cos B-(cos(A - B) + cos(A + B)] EXAMPLE 9 Evaluate sin 4r cos Sr de. SOLUTION This integral could be evaluated using integration by parts, but it's easier to use the identity in Equation 2(a) as follows: S sin dx con Sr dr - lsin-x) + sin 9.] da =!j(-sin x + sin 94) dx
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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