Explanation Given: The intgeral is ∫ d x 12 x − x 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: cos 2 x = 1 − sin 2 x Integration formula: ∫ 1 d x = x + c Calculation: The intgeral is ∫ d x 12 x − x 2 . We can rewrite the integral as: ∫ d x 12 x − x 2 = ∫ d x 36 − ( x − 6 ) 2 ... (1) Let x − 6 = 6 sin θ . Now, differentiate x − 6 = 6 sin θ . d x = 6 cos θ d θ Substitute 6 sin θ for x − 6 and 6 cos θ d θ for d x in equation (1)...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 8.3, Problem 38E

To determine

Evaluate the integral $\int \frac{d x}{\sqrt{12 x - x^{2}}}$ . First, complete the square to write $12 x - x^{2} = 36 - {(x - 6)}^{2}$ .

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Chapter 8.3, Problem 37E

Chapter 8.3, Problem 39E

Chapter 8 Solutions

CALCULUS (CLOTH)

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Evaluate the integral ∫ d x 12 x − x 2 . First, complete the square to write 12 x − x 2 = 36 − ( x − 6 ) 2 .

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

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Evaluate the integral $\int \frac{d x}{\sqrt{12 x - x^{2}}}$ . First, complete the square to write $12 x - x^{2} = 36 - {(x - 6)}^{2}$ .

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Evaluate the integral ∫ d x 12 x − x 2 . First, complete the square to write 12 x − x 2 = 36 − ( x − 6 ) 2 .

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

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Evaluate the integral ∫dx12x−x2. First, complete the square to write 12x−x2=36−(x−6)2.

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Chapter 8 Solutions

Evaluate the integral $\int \frac{d x}{\sqrt{12 x - x^{2}}}$ . First, complete the square to write $12 x - x^{2} = 36 - {(x - 6)}^{2}$ .