Explanation Given information: ∫ ( 1 − r 2 ) 5 / 2 r 8 d r Formula: cos 2 θ = 1 − sin 2 θ d d θ sin θ = cos θ d d θ cot θ = − csc 2 θ ∫ x n d x = x n + 1 n + 1 + C Calculation: Evaluate the integral as below. ∫ ( 1 − r 2 ) 5 / 2 r 8 d r (1) Apply trigonometric substitution by using the variable r as sin θ . r = sin θ Differentiate the term with respect to the variable θ as below d r d θ = cos θ d r = cos θ d θ Consider this term ( 1 − r 2 ) 5 / 2 by substituting sin θ for r . ( 1 − r 2 ) 5 / 2 = ( 1 − sin 2 θ ) 5 / 2 = ( cos 2 θ ) 5 / 2 = cos 5 θ Substitute cos 5 θ for ( 1 − r 2 ) 5 / 2 , sin θ for r and cos θ d θ for d r in equation (1). ∫ ( 1 − r 2 ) 5 / 2 r 8 d r = ∫ cos 5 θ cos θ sin 8 θ d θ = ∫ cos 6 θ sin 8 θ d θ = ∫ ( cos 6 θ sin 6 θ ) ( 1 sin 2 θ ) d θ ∫ ( 1 − r 2 ) 5 / 2 r 8 d r = ∫ cot 6 θ csc 2 θ d θ (2) Assume the variable u as cot θ

University Calculus: Early Transcendentals, Loose-leaf Edition (4th Edition)

4th Edition

ISBN: 9780135164860

Author: Joel R. Hass, Christopher Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 8.3, Problem 38E

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The value of the integral.

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