Explanation Given: The intgeral is ∫ d x 4 x 2 + 9 . 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: ∫ sec x d x = ln | sec x + tan x | + c Calculation: Part a: The intgeral is ∫ d x 4 x 2 + 9 . Let I = ∫ d x 4 x 2 + 9 . ... (1) Let x = 3 2 tan θ . Now, differentiate x = 3 2 tan θ . d x = 3 2 sec 2 θ d θ Substitute 3 2 tan θ for x and 3 2 sec 2 θ d θ for d x in equation (1). I = ∫ d x 4 x 2 + 9 = ∫ 3 2 sec 2 θ d θ 4 ( 3 2 tan θ ) 2 + 9 Simplify the above function...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 8.3, Problem 3E

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Evaluate the integral by the following steps.

a. Show that the substitution $x = \frac{3}{2} \tan θ$ transforms $I$ into $\frac{1}{2} \int \sec θ d θ$ .

b. Evaluate $I$ in terms of $θ$ .

c. Evaluate $I$ in term of $x$ .

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Chapter 8.3, Problem 2E

Chapter 8.3, Problem 4E

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1 -dx 2x + 4 Use SUBSTITUTION to evaluate the definite integral to three decimal places. Clearly demonstrate the following steps in order to earn full credit: a) What is "u"? What is "du"? b) Rewrite the integral in terms of u and du. c) Evaluate the integral. d) Evaluate the solution. Round final answer to 3 decimal places.

A, B and C please

dx x² +9 2. Use an appropriate trigonometric substitution to rewrite J as J(a simplified trig function of 0) d0. Carefully show your work with differentials and use of trig identities to help simplify. Then find the integral. Show any use of substitutions and reasoning with right triangles.

Chapter 8 Solutions

Calculus - Standalone book

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 the following steps. a. Show that the substitution x = 3 2 tan θ transforms I into 1 2 ∫ sec θ d θ . b. Evaluate I in terms of θ . c. Evaluate I in term of x .

Solution Summary: The author explains the steps of integration by substitution by showing that the substitution x = 3 2 tan transforms I into 1 2

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Evaluate the integral by the following steps.

a. Show that the substitution $x = \frac{3}{2} \tan θ$ transforms $I$ into $\frac{1}{2} \int \sec θ d θ$ .

b. Evaluate $I$ in terms of $θ$ .

c. Evaluate $I$ in term of $x$ .

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Chapter 8 Solutions

Evaluate the integral by the following steps. a. Show that the substitution x = 3 2 tan θ transforms I into 1 2 ∫ sec θ d θ . b. Evaluate I in terms of θ . c. Evaluate I in term of x .

Solution Summary: The author explains the steps of integration by substitution by showing that the substitution x = 3 2 tan transforms I into 1 2

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Evaluate the integral by the following steps. a. Show that the substitution x=32tanθ transforms I into 12∫secθ dθ. b. Evaluate I in terms of θ. c. Evaluate I in term of x.

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Chapter 8 Solutions

Evaluate the integral by the following steps.

a. Show that the substitution $x = \frac{3}{2} \tan θ$ transforms $I$ into $\frac{1}{2} \int \sec θ d θ$ .

b. Evaluate $I$ in terms of $θ$ .

c. Evaluate $I$ in term of $x$ .