Explanation Given: The intgeral is ∫ x 2 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. Trigonometic formula: tan 2 x = sec 2 x − 1 Integration formula: ∫ cosh x d x = sinh x + c Calculation: Integration by direct substitution: The intgeral is ∫ x 2 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 ⋅ x d x x 2 − 4 = ∫ x d u 2 u No, the substitution u = x 2 − 4 is not effective for evaluating the integral ∫ x 2 d x x 2 − 4 . Integration by trigonometric substitution: The intgeral is ∫ x 2 d x x 2 − 4 . ... (2) Let x = 2 sec θ . Now, differentiate x = 2 sec θ . d x = 2 sec θ tan θ d θ Taking square on both sides. x 2 = ( 2 sec θ ) 2 x 2 = 4 sec 2 θ Subtract 4 on both sides. x 2 − 4 = 4 sec 2 θ − 4 x 2 − 4 = 4 ( sec 2 θ − 1 ) x 2 − 4 = 4 tan 2 θ Substitute 2 sec θ for x and 2 sec θ tan θ d θ for d x in equation (2). ∫ x 2 d x x 2 − 4 = ∫ 4 sec 2 θ 2 sec θ tan θ d θ 4 tan 2 θ Simplify the above function...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.3, Problem 12E

To determine

Check the substitution $u = x^{2} - 4$ effective for solving the integral $\int \frac{x^{2} d x}{\sqrt{x^{2} - 4}}$ ? If not, evaluate by using trigonometric substitution.

Expert Solution & Answer

Chapter 8 Solutions

CALCULUS (CLOTH)

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Check the substitution u = x 2 − 4 effective for solving the integral ∫ x 2 d x x 2 − 4 ? If not, evaluate by using trigonometric substitution.

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

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Check the substitution $u = x^{2} - 4$ effective for solving the integral $\int \frac{x^{2} d x}{\sqrt{x^{2} - 4}}$ ? If not, evaluate by using trigonometric substitution.

Chapter 8 Solutions

Check the substitution u = x 2 − 4 effective for solving the integral ∫ x 2 d x x 2 − 4 ? If not, evaluate by using trigonometric substitution.

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

Videos

Check the substitution u=x2−4 effective for solving the integral ∫x2 dxx2−4 ? If not, evaluate by using trigonometric substitution.

Chapter 8 Solutions

Check the substitution $u = x^{2} - 4$ effective for solving the integral $\int \frac{x^{2} d x}{\sqrt{x^{2} - 4}}$ ? If not, evaluate by using trigonometric substitution.