Explanation Given: The intgeral is ∫ 16 − 5 x 2 d x , x = 4 5 sin θ . 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. Formula: ∫ cos x d x = sin x + c Calculation: The intgeral is ∫ 16 − 5 x 2 d x , x = 4 5 sin θ . Let I = ∫ 16 − 5 x 2 d x . ... (1) Let x = 4 5 sin θ . Now, differentiate x = 4 5 sin θ . d x = 4 5 cos θ d θ Substitute 4 5 sin θ for x and 4 5 cos θ d θ for d x in equation (1). I = ∫ 16 − 5 x 2 d x = ∫ 16 − 5 ( 4 5 sin θ ) 2 4 5 cos θ d θ Simplify the above function. ∫ 16 − 5 ( 4 5 sin θ ) 2 4 5 cos θ d θ = ∫ 16 − 16 sin 2 θ 4 5 cos θ d θ = ∫ 4 1 − sin 2 θ 4 5 cos θ d θ = 4 × 4 5 ∫ cos θ cos θ d θ = 16 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 5E

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Evaluate the integral by using indicated substitution.

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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 using indicated substitution.

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Evaluate the integral by using indicated substitution.

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