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a) 27 " dᎾ 5 + 4 sin 0

a) 27 " dᎾ 5 + 4 sin 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.2: Trigonometric Functions Of Angles
Problem 32E
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(2)-The question is about Calculus of Residues. (Integration of functions using complex functions).

z is a complex number.

One of the photos shows the solution. Use only the given solution.

Solve parts a, b and c.

21.5. Evaluate each of the following integrals by turning them into contour integrals around the unit circle. 21.5 Problems 537 27T 2T do do (a) (b) where a > 1. 5 + 4 sin 0 a + cos 0 27T do do (c) (d) where a, b > 0. 1+ sin? 0 (a + b cos? 0)²
Transcribed Image Text:21.5. Evaluate each of the following integrals by turning them into contour integrals around the unit circle. 21.5 Problems 537 27T 2T do do (a) (b) where a > 1. 5 + 4 sin 0 a + cos 0 27T do do (c) (d) where a, b > 0. 1+ sin? 0 (a + b cos? 0)²
21.4 Functions of Trigonometric Functions The third typc of integral wc can cvaluate using the residuc theorcm involvcs only trigonometric functions and is typically of the form F(sin 0, cos 0) d0, where F is some (typically rational) function³ of its arguments. Since 0 varies from 0 to 27, we can consider it as the angle of a point z on the unit circle 3 Recall that a rational function is, by definition, the ratio of two polynomials. 21.4 Functions of Trigonometric Functions 535 centered at the origin. Then z = eit and e-io = 1/z, and we can substitute cos 8 = (z + 1/z)/2, sin 0 = (z – 1/2)/(2i), and dO = dz/(iz) in the original integral to obtain 1/z z +1/z iz dz F 2i C This integral can often be evaluated using the method of residues.
Transcribed Image Text:21.4 Functions of Trigonometric Functions The third typc of integral wc can cvaluate using the residuc theorcm involvcs only trigonometric functions and is typically of the form F(sin 0, cos 0) d0, where F is some (typically rational) function³ of its arguments. Since 0 varies from 0 to 27, we can consider it as the angle of a point z on the unit circle 3 Recall that a rational function is, by definition, the ratio of two polynomials. 21.4 Functions of Trigonometric Functions 535 centered at the origin. Then z = eit and e-io = 1/z, and we can substitute cos 8 = (z + 1/z)/2, sin 0 = (z – 1/2)/(2i), and dO = dz/(iz) in the original integral to obtain 1/z z +1/z iz dz F 2i C This integral can often be evaluated using the method of residues.
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