2. Using the substitution 0 = 2 arctan t, it is possible to transform any rational function of trig functions into a rational function of t, and hence use partial fractions. In this problem, you will use this substitution to (almost) find an antiderivative of 1 cos + sin 0 (a) Using the double-angle identity cos(24) = 2 (cos p)2-1, and a triangle representing tan y = 1, it is possible to find that cos 0 = 2 2 (√²+²)²³ - 1 = 1 + ²²* Find a similar formula for sin 0. 1-t² (b) Find a formula for de. (c) Plug the results of (a) and (b) in appropriately to find a new form of the integral which does not make any reference to trigonometric functions. Simplify the integrand completely.
2. Using the substitution 0 = 2 arctan t, it is possible to transform any rational function of trig functions into a rational function of t, and hence use partial fractions. In this problem, you will use this substitution to (almost) find an antiderivative of 1 cos + sin 0 (a) Using the double-angle identity cos(24) = 2 (cos p)2-1, and a triangle representing tan y = 1, it is possible to find that cos 0 = 2 2 (√²+²)²³ - 1 = 1 + ²²* Find a similar formula for sin 0. 1-t² (b) Find a formula for de. (c) Plug the results of (a) and (b) in appropriately to find a new form of the integral which does not make any reference to trigonometric functions. Simplify the integrand completely.
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:alc 2, Fall 2022
roblem Set 2
2. Using the substitution 0 = 2 arctan t, it is possible to transform any rational function of trig
functions into a rational function of t, and hence use partial fractions. In this problem, you
will use this substitution to (almost) find an antiderivative of
(a) Using the double-angle identity cos(24) = 2 (cos y)²-1, and a triangle representing tan y = 1,
1-t²
it is possible to find that cos 0 = 2(
Find a similar formula for sin 0.
2 ( ₁₁² ) ² − 1 = 1 +²²²
√1+1²
(b) Find a formula for de.
1
cos + sin 0
(c) Plug the results of (a) and (b) in appropriately to find a new form of the integral which does
not make any reference to trigonometric functions. Simplify the integrand completely.
8:
H
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