1. This question is designed to develop an understanding for integrals of the form Scos(a) dx for positive integers n. A. Deduce that f da is equal to both In 1-sin(2) + C and In sec(x) + tan(x)| + C. (Hint: 1+sin(x) 1-sin(x) (1+sin(x))² 1-sin²(x) :.) cos(x) B. Deduce that cos²¹(a) dx is equal to tan(x) + C. C. Let n be a positive integer at least 3 in value, and let In = cos(x) dx. Use integration by parts to obtain that In = In-2 + sin(x) cos¹-n (x). (Hint: Use the identity 1 = cos² (x) + sin² (x).) D. Your results from Parts A, B, and C logically combine to allow us to integrate any function of the form f dx for positive integers n. Explain. cos" (x)
1. This question is designed to develop an understanding for integrals of the form Scos(a) dx for positive integers n. A. Deduce that f da is equal to both In 1-sin(2) + C and In sec(x) + tan(x)| + C. (Hint: 1+sin(x) 1-sin(x) (1+sin(x))² 1-sin²(x) :.) cos(x) B. Deduce that cos²¹(a) dx is equal to tan(x) + C. C. Let n be a positive integer at least 3 in value, and let In = cos(x) dx. Use integration by parts to obtain that In = In-2 + sin(x) cos¹-n (x). (Hint: Use the identity 1 = cos² (x) + sin² (x).) D. Your results from Parts A, B, and C logically combine to allow us to integrate any function of the form f dx for positive integers n. Explain. cos" (x)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. This question is designed to develop an understanding for integrals of the form ſ dx for positive integers n.
Cosn
+ C and In|sec(x) + tan(x)| + C. (Hint:
A. Deduce that f
B. Deduce that f
1
cos(x)
1+sin(x)
1-sin(x)
dx is equal to both
dx is equal to tan(x) + C.
cos² (x)
C. Let n be a positive integer at least 3 in value, and let In
1
cos (x)
n-2
1
In = n=1 In-2 + ² sin(x) cos¹-(x). (Hint: Use the identity 1 = cos² (x) + sin² (x).)
-1
n-1
1
D. Your results from Parts A, B, and C logically combine to allow us to integrate any function of the form f dx for positive integers n. Explain.
cos" (x)
=
1+sin(x) (1+sin(x))²
1-sin(x) 1-sin²(x)
S
-
dx. Use integration by parts to obtain that
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