Explanation Given: The integral is ∫ d x x 2 − 1 . 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. Integration formula: ∫ csc x d x = ln | csc x − cot x | + c Calculation: The integral is ∫ d x x 2 − 1 . (1) Let x = sec θ . Differentiate x = sec θ . d x = sec θ tan θ d θ Substitute sec θ for x and sec θ tan θ d θ for d x in equation (1). ∫ d x x 2 − 1 = ∫ sec θ tan θ d θ ( sec θ ) 2 − 1 = ∫ sec θ tan θ d θ sec 2 θ − 1 = ∫ sec θ tan θ d θ tan 2 θ = ∫ sec θ d θ tan θ = ∫ d θ sin θ = ∫ csc θ d θ Integrate the function. ∫ csc θ d θ = ln | csc θ − cot θ | + c Find the value of csc θ ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Chapter 8.3, Problem 53E

To determine

Evaluate the integral $\int \frac{d x}{x^{2} - 1}$ in two ways and check the answer will same via trigonometric substitution and then using the identity $\frac{1}{x^{2} - 1} = \frac{1}{2} (\frac{1}{x - 1} - \frac{1}{x + 1})$ .

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Chapter 8.3, Problem 52E

Chapter 8.3, Problem 54E

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Chapter 8 Solutions

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Evaluate the integral ∫ d x x 2 − 1 in two ways and check the answer will same via trigonometric substitution and then using the identity 1 x 2 − 1 = 1 2 ( 1 x − 1 − 1 x + 1 ) .

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Evaluate the integral $\int \frac{d x}{x^{2} - 1}$ in two ways and check the answer will same via trigonometric substitution and then using the identity $\frac{1}{x^{2} - 1} = \frac{1}{2} (\frac{1}{x - 1} - \frac{1}{x + 1})$ .

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Evaluate the integral ∫ d x x 2 − 1 in two ways and check the answer will same via trigonometric substitution and then using the identity 1 x 2 − 1 = 1 2 ( 1 x − 1 − 1 x + 1 ) .

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Evaluate the integral ∫dxx2−1 in two ways and check the answer will same via trigonometric substitution and then using the identity 1x2−1=12(1x−1−1x+1).

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Evaluate the integral $\int \frac{d x}{x^{2} - 1}$ in two ways and check the answer will same via trigonometric substitution and then using the identity $\frac{1}{x^{2} - 1} = \frac{1}{2} (\frac{1}{x - 1} - \frac{1}{x + 1})$ .