Explanation Given: The intgeral is ∫ d x ( x 2 + 6 x + 6 ) 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: tan 2 x = sec 2 x − 1 Integration formula: ∫ sec x d x = ln | sec x + tan x | + c Calculation: The intgeral is ∫ d x ( x 2 + 6 x + 6 ) 2 . First, make the completing square. ∫ d x ( x 2 + 6 x + 6 ) 2 = ∫ d x ( x 2 + 6 x + 9 − 3 ) 2 = ∫ d x ( ( x 2 + 6 x + 9 ) − 3 ) 2 = ∫ d x ( ( x + 3 ) 2 − 3 ) 2 ... (1) Let u = x + 3 . Now, differentiate u = x + 3 . d u = d x Substitute u for x + 3 and d u for d x in equation (1). ∫ d x ( ( x + 3 ) 2 − 3 ) 2 = ∫ d u ( u 2 − 3 ) 2 … (2) Now, again let u = 3 sec θ . Now, differentiate u = 3 sec θ . d u = 3 sec θ tan θ d θ Substitute 3 sec θ for u and 3 sec θ tan θ d θ for d u in equation (2). ∫ d u ( u 2 − 3 ) 2 = ∫ 3 sec θ tan θ d θ ( ( 3 sec θ ) 2 − 3 ) 2 Simplify the above function. ∫ 3 sec θ tan θ d θ ( ( 3 sec θ ) 2 − 3 ) 2 = ∫ 3 sec θ tan θ d θ ( 3 sec 2 θ − 3 ) 2 = ∫ 3 sec θ tan θ d θ 9 ( sec 2 θ − 1 ) 2 = ∫ 3 sec θ tan θ d θ 9 tan 4 θ = ∫ 3 sec θ d θ 9 tan 3 θ Simplify the above integral sec x = 1 cos x and tan x = sin x cos x . ∫ 3 sec θ d θ 9 tan 3 θ = 3 9 ∫ cos 2 θ d θ sin 3 θ = 3 9 ∫ 1 − sin 2 θ d θ sin 3 θ = 3 9 ∫ ( 1 sin 3 θ − sin 2 θ sin 3 θ ) d θ = 3 9 ∫ ( csc 3 θ − csc θ ) d θ Integrate the function...

4th Edition

ISBN: 9781319434540

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 8.3, Problem 44E

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Evaluate the integral by completing the square. Also, using trigonometric substitution.

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Chapter 8.3, Problem 43E

Chapter 8.3, Problem 45E

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Chapter 8 Solutions

ACHIEVE STANDALONE ACCESS F/ CALC 4E

Ch. 8.1 - Prob. 1PQ Ch. 8.1 - Prob. 2PQ Ch. 8.1 - Prob. 3PQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E

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Evaluate the integral by completing the square. Also, using trigonometric substitution.

Solution Summary: The author explains how to determine the integral by completing the square and using trigonometric substitution.

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Evaluate the integral by completing the square. Also, using trigonometric substitution.

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Chapter 8 Solutions