Explanation Given: The intgeral is ∫ x 2 d x ( x 2 + 1 ) 3 2 . Key concept used: Steps of integration by substitution: If the integral is in the below form: ∫ f ( g ( x ) ) g ′ ( x ) d x Let u = g ( x ) and d u = g ′ ( x ) d x The intgeral becomes, ∫ f ( u ) d u . Thus, integrate f ( u ) . After that substitute g ( x ) back. Trigonometic formula: sec 2 x = tan 2 x + 1 , tan x = sin x cos x Integration formula: ∫ cos x d x = sin x + c and ∫ sec x d x = ln | sec x + tan x | + c Calculation: The intgeral is ∫ x 2 d x ( x 2 + 1 ) 3 2 . ... (1) Let x = tan θ . Now, differentiate x = tan θ . d x = sec 2 θ d θ Substitute tan θ for x and sec 2 θ d θ for d x in equation (1). ∫ x 2 d x ( x 2 + 1 ) 3 2 = ∫ tan 2 θ sec 2 θ d θ ( tan 2 θ + 1 ) 3 2 Simplify the above function. ∫ tan 2 θ sec 2 θ d θ ( tan 2 θ + 1 ) 3 2 = ∫ tan 2 θ sec 2 θ d θ ( sec 2 θ ) 3 2 = ∫ tan 2 θ sec 2 θ d θ sec 3 θ = ∫ ( sec 2 θ − 1 ) d θ sec θ

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 32E

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Evaluate the integral using trigonometric substitution.

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Evaluate the integral using trigonometric substitution.

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Evaluate the integral using trigonometric substitution.

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