In like manner inverse hyperbolic functions are

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Applied Calculus for the Managerial, Life, and Social Sciences
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Chapter 8 / Exercise 47
Applied Calculus for the Managerial, Life, and Social Sciences
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Hyperbolic Functions are expressible in terms of exponential functions. In like manner, inverse hyperbolic functions are expressible in terms of natural logarithms. Example 14.2 Evaluate the following integrals: a) 16 x 9 dx 2 b) x cos 4 xdx cos 2 c) 5 x 6 x xdx 3 2 4 d) 3 2 2 5 x 12 x 9 dx Solutions: a) 16 x 9 dx 2 2 2 2 x 3 4 dx 3 3 1 x 9 16 dx a = 4 , u = 3x 4 x 3 , C 4 x 3 coth 4 1 4 x 3 , C 4 x 3 tanh 4 1 3 1 1 1   C x 3 4 x 3 4 ln 4 2 1 3 1 C x 3 4 x 3 4 ln 24 1
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Applied Calculus for the Managerial, Life, and Social Sciences
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Chapter 8 / Exercise 47
Applied Calculus for the Managerial, Life, and Social Sciences
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b) x cos 4 xdx cos 2 x sin 1 4 xdx cos 2 2 2 x sin 3 xdx cos a = 3 , u = sin x C 3 x cos 4 x sin ln C 3 x sin 3 x sin ln C 3 x sin sinh 2 2 1 c) 5 x 6 x xdx 3 2 4 Complete the square: 2 2 2 2 4 2 4 2 3 x 5 9 9 x 6 x 5 x 6 x C 2 5 x 6 x 3 x ln 2 3 C 2 4 3 x 3 x ln 2 3 C 2 3 x cosh 2 3 2 3 x xdx 2 2 3 2 4 2 2 2 2 2 1 2 2 2 d) 3 2 2 5 x 12 x 9 dx Complete the square: Use: 2 2 2 b a b ab 2 a 2 2 2 2 3 2 x 3 5 4 4 2 x 3 2 x 3 5 x 12 x 9 3 2 2 2 3 2 x 3 dx 3 3 1
3 2 1 3 2 x 3 cosh 3 1 3 2 2 3 9 2 x 3 2 x 3 ln 3 1 7 4 10 2 7 ln 3 1 7 4 10 4 7 ln 3 1 3 9 16 4 3 9 49 7 ln 3 1
EXERCISES: 2. Evaluate the following integrals using the integration yielding to inverse hyperbolic functions: a. 2 x 4 dx e. 4 16 x xdx b. 2 x x 4 2 dx f. 3 4 2 t t dt c. 1 x xdx 4 g. 1 0 2 2 2 z z dz d. 16 e 9 dx e x 4 x 2 h. 2 / 0 2 sin 1 cos d 3. Find the area between the curve y = Sinh -1 x and the x-axis from x = 0 to x =2. 4. Find the area between the curve y = Cosh -1 x and the line x = 4.
MODULE IV TECHNIQUES OF INTEGRATION CONTENTS: Lesson 15: Integration by Parts Lesson 16: Integration by Trigonometric Substitution Lesson 17: The Miscellaneous Substitution Lesson 18: Integration by Partial Fractions OVERVIEW OF THE MODULE: In this module, we introduce the techniques of integration to evaluate variety of integrals which an ordinary integration formula could not evaluate. In each of the technique used, we consider the basic integration formulas we derived from the previous topics discussed. Each of these techniques is a means of transforming integrals into forms that can be evaluated using the fundamentals of integration. These techniques of integration include integration by parts, integration by trigonometric substitution, integration of rational function of sine and cosine and the miscellaneous substitution.

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