Explanation Given: The intgeral is ∫ x d x x 2 − 4 . 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: ∫ cosh x d x = sinh x + c Calculation: Integration by direct substitution: The intgeral is ∫ x d x x 2 − 4 . ... (1) Let u = x 2 − 4 . Now, differentiate u = x 2 − 4 . d u = 2 x d x d u 2 = x d x Substitute u for x 2 − 4 and d u 2 for x d x in equation (1). ∫ x d x x 2 − 4 = ∫ d u 2 u Simplify the above function. ∫ d u 2 u = 1 2 ∫ u − 1 2 d u Integrate the function. 1 2 ∫ u − 1 2 d u = 1 2 ⋅ u − 1 2 + 1 − 1 2 + 1 + c = 1 2 ⋅ u 1 2 1 2 + c = u + c Now, substitute the value back u = x 2 − 4 . u + c = x 2 − 4 + c Thus, the value of ∫ x d x x 2 − 4 is x 2 − 4 + c . Integration by trigonometric substitution: The intgeral is ∫ x d x x 2 − 4 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 11E

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Evaluate the integral using the direct substitution $u = x^{2} - 4$ and by trigonometric substitution.

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Chapter 8.3, Problem 10E

Chapter 8.3, Problem 12E

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Evaluate the integral using the direct substitution u = x 2 − 4 and by trigonometric substitution.

Solution Summary: The author explains the steps of integration by direct substitution and trigonometric substitution.

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Evaluate the integral using the direct substitution $u = x^{2} - 4$ and by trigonometric substitution.

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Chapter 8 Solutions

Evaluate the integral using the direct substitution u = x 2 − 4 and by trigonometric substitution.

Solution Summary: The author explains the steps of integration by direct substitution and trigonometric substitution.

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Evaluate the integral using the direct substitution u=x2−4 and by trigonometric substitution.

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Chapter 8 Solutions

Evaluate the integral using the direct substitution $u = x^{2} - 4$ and by trigonometric substitution.