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1. In this problem, we will consider two methods that will allow us to evaluate integrals of the form sin(ar) cos(bar) d.r, | cos(ax) cos(bar) dr, or sin(ar) sin(bar) dz, where a + b. The first method will rely on the trigonometric identities cos(r + y) = cos r cos y + sin z sin y and sin(r ± y) = sin z cos y ± sin y cos r, while the second method will use integration by parts. (a) Use the above identities to find a formula for cos(72) that involves sines and cosines of the angles 2x and 5x. (b) Use the above identities to find a formula for cos(3r) that involves sines and cosines of the angles 2x and 5x. (c) Use the results from parts (a) and (b) to find a formula for cos(2r) cos(52) in terms of cos(3r) and cos(7r). (d) Use your formula from part (c) to help you evaluate / cos(2r) cos(5æ) dr. (e) Use integration by parts (twice) to evaluate / cos(2r) cos(5æ) dr. (f) Show that the results in parts (d) and (e) differ by a constant. (Hint: some trigonometric identities could be very helpful!)

1. In this problem, we will consider two methods that will allow us to evaluate integrals of the form sin(ar) cos(bar) d.r, | cos(ax) cos(bar) dr, or sin(ar) sin(bar) dz, where a + b. The first method will rely on the trigonometric identities cos(r + y) = cos r cos y + sin z sin y and sin(r ± y) = sin z cos y ± sin y cos r, while the second method will use integration by parts. (a) Use the above identities to find a formula for cos(72) that involves sines and cosines of the angles 2x and 5x. (b) Use the above identities to find a formula for cos(3r) that involves sines and cosines of the angles 2x and 5x. (c) Use the results from parts (a) and (b) to find a formula for cos(2r) cos(52) in terms of cos(3r) and cos(7r). (d) Use your formula from part (c) to help you evaluate / cos(2r) cos(5æ) dr. (e) Use integration by parts (twice) to evaluate / cos(2r) cos(5æ) dr. (f) Show that the results in parts (d) and (e) differ by a constant. (Hint: some trigonometric identities could be very helpful!)

Calculus: Early Transcendentals
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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1. In this problem, we will consider two methods that will allow us to evaluate integrals of the form sin(ar) cos(bar) d.r, | cost cos(a.r) cos(bar) dr, or / sin(ar) sin(br) dr, where a + b. The first method will rely on the trigonometric identities cos(r + y) = cos r cos y + sin a sin y and sin(r +y) = sin a cos y ± sin y cos a, while the second method will use integration by parts. (a) Use the above identities to find a formula for cos(7x) that involves sines and cosines of the angles 2:r and 5x. (b) Use the above identities to find a formula for cos(3r) that involves sines and cosines of the angles 2r and 5x. (c) Use the results from parts (a) and (b) to find a formula for cos(2r) cos(5æ) in terms of cos(3r) and Cos(7r). (d) Use your formula from part (c) to help you evaluate | cos(2r) cos(5.r) da. (e) Use integration by parts (twice) to evaluate / cos(2r) cos(5.r) d.r. (f) Show that the results in parts (d) and (e) differ by a constant. (Hint: some trigonometric identities could be very helpful!)
Transcribed Image Text:1. In this problem, we will consider two methods that will allow us to evaluate integrals of the form sin(ar) cos(bar) d.r, | cost cos(a.r) cos(bar) dr, or / sin(ar) sin(br) dr, where a + b. The first method will rely on the trigonometric identities cos(r + y) = cos r cos y + sin a sin y and sin(r +y) = sin a cos y ± sin y cos a, while the second method will use integration by parts. (a) Use the above identities to find a formula for cos(7x) that involves sines and cosines of the angles 2:r and 5x. (b) Use the above identities to find a formula for cos(3r) that involves sines and cosines of the angles 2r and 5x. (c) Use the results from parts (a) and (b) to find a formula for cos(2r) cos(5æ) in terms of cos(3r) and Cos(7r). (d) Use your formula from part (c) to help you evaluate | cos(2r) cos(5.r) da. (e) Use integration by parts (twice) to evaluate / cos(2r) cos(5.r) d.r. (f) Show that the results in parts (d) and (e) differ by a constant. (Hint: some trigonometric identities could be very helpful!)
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