# 1 sinh x 3 4 2 sinh x 4 3 3 cosh x 1 1 7 5 x 4 cosh x

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Chapter 5 / Exercise 14
Algebra and Trigonometry
Redlin/Stewart Expert Verified
1. sinh x 3 4 2. sinh x 4 3 3. cosh x 1 1 7 5 , x 0 4. cosh x 1 5 3 , x 0 In Exercises 5–10, rewrite the expression in terms of exponentials and simplify the results as much as you can. Support your answers graphically. 5. 2 cosh ln x 6. sinh 2 ln x 7. cosh 5 x sinh 5 x 8. cosh 3 x sinh 3 x 9. sinh x cosh x 4 10. ln cosh x sinh x ln cosh x sinh x 11. Use the identities sinh x y sinh x cosh y cosh x sinh y cosh x y cosh x cosh y sinh x sinh y to show that (a) sinh 2 x 2 sinh x cosh x ; (b) cosh 2 x cosh 2 x sinh 2 x . 12. Use the definitions of cosh x and sinh x to show that cosh 2 x sinh 2 x 1. In Exercises 13–24, find the derivative of y with respect to the appropriate variable. 13. y 6 sinh 3 x 14. y 1 2 sinh 2 x 1 15. y 2 t tanh t 16. y t 2 tanh 1 t 17. y ln sinh z 18. y ln cosh z EXAMPLE 4 Using Table A6.6 Evaluate 1 0 3 2 dx 4 x 2 . SOLUTION Solve Analytically The indefinite integral is 3 2 dx 4 x 2 a 2 du u 2 u 2 x , du 2 dx , a 3 sinh 1 ( a u ) C Formula from Table A6.6 sinh 1 ( 2 x 3 ) C. Therefore, 1 0 3 2 dx 4 x 2 sinh 1 ( 2 x 3 ) ] 1 0 sinh 1 ( 2 3 ) sinh 1 0 sinh 1 ( 2 3 ) 0 0.98665. Support Numerically To five decimal places, NINT 2 3 4 x 2 , x , 0, 1 0.98665. Now try Exercise 37. Section A6 Exercises
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Chapter 5 / Exercise 14
Algebra and Trigonometry
Redlin/Stewart Expert Verified
Section A6 Hyperbolic Functions 609 19. y sech u 1 ln sech u 20. y csch u 1 ln csch u 21. y ln cosh x 1 2 tanh 2 x 22. y ln sinh x 1 2 coth 2 x 23. y x 2 1 sech ln x ( Hint: Before differentiating, express in terms of exponentials and simplify.) 24. y 4 x 2 1 csch ln 2 x In Exercises 25–36, find the derivative of y with respect to the appropriate variable. 25. y sinh 1 x 26. y cosh 1 2 x 1 27. y 1 u tanh 1 u 28. y u 2 2 u tanh 1 u 1 29. y 1 t coth 1 t 30. y 1 t 2 coth 1 t 31. y cos 1 x x sech 1 x 32. y ln x 1 x 2 sech 1 x 33. y csch 1 ( 1 2 ) u 34. y csch 1 2 u 35. y sinh 1 tan x 36. y cosh 1 sec x , 0 x p 2 Verify the integration formulas in Exercises 37–40. 37. (a) sech x dx tan 1 sinh x C (b) sech x dx sin 1 tanh x C 38. x sech 1 x dx x 2 2 sech 1 x 1 2 1 x 2 C 39. x coth 1 x dx x 2 2 1 coth 1 x 2 x C 40. tanh 1 x dx x tanh 1 x 1 2 ln 1 x 2 C Evaluate the integrals in Exercises 41–50. 41. sinh 2 x dx 42. sinh 5 x dx 43. 6 cosh ( 2 x ln 3 ) dx 44. 4 cosh 3 x ln 2 dx 45. tanh 7 x dx 46. coth u 3 d u 47. sech 2 ( x 1 2 ) dx 48. csch 2 5 x dx 49. 50. Evaluate the integrals in Exercises 51–60 analytically and support with NINT. 51. ln 4 ln 2 coth x dx 52. ln 2 0 tanh 2 x dx 53. ln 2 ln 4 2 e u cosh u d u 54. ln 2 0 4 e u sinh u d u 55. p /4 p /4 cosh tan u sec 2 u d u 56. p 2 0 2 sinh sin u cos u d u 57. 2 1 cosh t ln t dt 58. 4 1 8 cosh x x dx 59. 0 ln 2 cosh 2 ( 2 x ) dx 60. ln 10 0 4 sinh 2 ( 2 x ) dx In Exercises 61 and 62, find the volume of the solid generated by revolving the shaded region about the x -axis. 61. 62. –ln 3 x y 1 ln 3 0 y sech x x y 1 1 0 y cosh x 2 2 3 y sinh x csch ln t coth ln t dt t sech t tanh t dt t
610 Appendices 63. Find the volume of the solid generated by revolving the shaded region about the line y 1.
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