SalasSV_08_06_ex - . 39. (a) Use the approach given in...

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8.6 SOME RATIONALIZING SUBSTITUTIONS ± 485 EXERCISES *8.6 Calculate the integral. 1. ± dx 1 x . 2. ± x 1 + x dx . 3. ± 1 + e x dx . 4. ± dx x ( x 1 / 3 1) . 5. ± x 1 + xdx . [(a) set u 2 = 1 + x ; (b) set u = 1 + x ] 6. ± x 2 1 + xdx . [(a) set u 2 = 1 + x ; (b) set u = 1 + x ] 7. ± ( x + 2) x 1 dx . 8. ± ( x 1) x + 2 dx . 9. ± x 3 (1 + x 2 ) 3 dx . 10. ± x (1 + x ) 1 / 3 dx . 11. ± x x 1 dx . 12. ± x x + 1 dx . 13. ± x 1 + 1 x 1 1 dx . 14. ± 1 e x 1 + e x dx . 15. ± dx 1 + e x . 16. ± dx 1 + e x . 17. ± x x + 4 dx . 18. ± x + 1 x x 2 dx . 19. ± 2 x 2 (4 x + 1) 5 / 2 dx . 20. ± x 2 x 1 dx . 21. ± x ( ax + b ) 3 / 2 dx . 22. ± x ax + b dx . 23. ± 1 1 + cos x sin x dx . 24. ± 1 2 + cos x dx . 25. ± 1 2 + sin x dx . 26. ± sin x 1 + sin 2 x dx . 27. ± 1 sin x + tan x dx . 28. ± 1 1 + sin x + cos x dx . 29. ± 1 cos x 1 + sin x dx . 30. ± 1 5 + 3 sin x dx . Evaluate the defnite integral. 31. ± 4 0 x 3 / 2 x + 1 dx . 32. ± 8 0 1 1 + 3 x dx . 33. ± π/ 2 0 sin 2 x 2 + cos x dx . 34. ± π/ 2 0 1 1 + sin x dx . 35. ± π/ 3 0 1 sin x cos x 1 dx . 36. ± 1 0 x 1 + x dx . 37. Use the method oF this section to show that ± sec xdx = ± 1 cos x dx = ln ² ² ² ² 1 + tan ( x / 2) 1 tan ( x / 2) ² ² ² ² + C . 38. (a) Another way to calculate ³ sec xdx is to write ± sec xdx = ± cos x cos 2 x dx = ± cos x 1 sin 2 x dx . Use the method oF this section to show that ± sec xdx = ln ´ 1 + sin x 1 sin x + C . (b) Show that the result oF part (a) is equivalent to the Familiar Formula ± sec xdx = ln | sec x + tan x |+ C
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Unformatted text preview: . 39. (a) Use the approach given in Exercise 38(a) to show that ± csc x dx = ln ´ 1 − cos x 1 + cos x + C . (b) Show that the result oF part (a) is equivalent to the Formula ± csc x dx = ln | csc x − cot x | + C . 40. The integral oF a rational Function oF sinh x and cosh x can be transFormed into a rational Function oF u by means oF the substitution u = tanh ( x / 2). Show that this substitution, gives sinh x = 2 u 1 − u 2 , cosh x = 1 + u 2 1 − u 2 , dx = 2 1 − u 2 du . In Exercises 41–44, set u = tanh ( x / 2) and carry out the integration 41. ± sech x dx . 42. ± 1 1 + cosh x dx . 43. ± 1 sinh x + cosh x dx . 44. ± 1 − e x 1 + e x dx ....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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