EXAMPLE 9 Since f(x) = (tan x)/ (1 + x2 + x4) satisfies f(-x) = -f(x), it is odd and so tan x -1+12 + , 4 dx =0 4.5 EXERCISES 1-6 Evaluate the integral by making the given substitution. dt 23. /1+ 23 -dz 24. cost 1 + tant 1. cos 2x dx, u = 2x 2 . x ( 2 x 2 + 3 ) # dx , 1 = 2 x 2 + 3 25. cot x csc'x dx 26. dx tan2x 3. [x' V x3 + 1 dx , 1 = x3 + 1 27 . sec'x tan x dx 28 . x 2 2 + x dx 4. | sin '0 cos0 do, u = sine 29 . * ( 2x + 5 ) 8 dx 30 . [x 3 V x 2 + 1 dx 5 . ( x4 - 5) 2 dx , 1 = 14 - 5 31-34 Evaluate the indefinite integral. Illustrate and check that 6. V 2t + 1 dt , u = 2+ + 1 your answer is reasonable by graphing both the function and its antiderivative (take C = 0). 31 . (x ( x 2 - 1 ) 3 dx 32 . tan '0 sec 2 0 do 7-30 Evaluate the indefinite integral. 7. x V 1 - x2 dx 8. [x2 sin(x 3 ) dx 33 . sin'x cos x dx 34 . sin x cos*x dx 9 . ( 1 - 2x ) dx 10 . sint VI + cost dt 35-51 Evaluate the definite integral. 11. sin ( 20/ 3 ) de 12 . sec 2 20 do 35. COS ( Tr t / 2 ) dt 36 . ( 3t - 1 ) 50 dt 13 . sec 3t tan 3t dt 14 . ) 2 ( 4 - 1 3 ) 213 dy 37. V1 + 7x dx 38 . " x cos ( x ? ) dx 15. cos ( 1 + 5t) dt 16. ( sin Vx dx 39, 7/6 sin t Vx cost dt 40. Ja./3 CSC2 ( 2t) dt 17 . sec -0 tan '0 de 18 . sin x sin(cos x) dx _ma ( x3 + x* tan x) dx 42. [ cos x sin(sin x) dx 19 . ( x 2 + 1 ) ( * 3 + 3x ) + dx 20 . [x x + 2 dx 43 . [13 - dx V ( 1 + 2 x ) 2 44 . " x Vaz - x 2 dx 21. a + bx 2 = dx COS(TT / x) 3ax + bx3 22. -dx x 2 45. " x x2 + a? dx ( a > 0) 46. " x4 sin x dxthis equation geomet 50. sin(271 / T - a) dt 63. If a and b are positiv dx 57. Jo ( 1 + Vx )' 64. If f is continuous on 52. Verify that f(x) = sin Vx is an odd function and use that to show that fact to show that J." xf (sin o, sin Vidx 1 65. If f is continuous, pro 53-54 Use a graph to give a rough estimate of the area of the region that lies under the given curve. Then find the exact area. f(cos x 53. y = V2x + 1, Ox=1 54. y = 2 sin x - sin 2x, OS X T 66. Use Exercise 65 to eva 55. Evaluate 2, (x + 3)v4 - x2 dx by writing it as a sum of The following exercises are two integrals and interpreting one of those integrals in terms already covered Chapter 6. of an area. 67-84 Evaluate the integral 56. Evaluate foxv1 - x4 dx by making a substitution and interpreting the resulting integral in terms of an area. dx 67. 5-3x 57. Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes 69 . . (In x ) 2 about 5 s. The maximum rate of air flow into the lungs x " dx is about 0.5 L/s. This explains, in part, why the function f(t) = 2 sin(27r t/5) has often been used to model the rate 71 . e VItex dx of air flow into the lungs. Use this model to find the volume of inhaled air in the lungs at time t. 73 . (arctan x)2 - dx x2+ 1 58. A model for the basal metabolism rate, in kcal/h, of a young man is R(t) = 85 - 0.18 cos( 7r t/12), where t is the 1+x time in hours measured from 5:00 AM. What is this man's 2 dx total basal metabolism, Jo R(t) dt, over a 24-hour time period? 77. sin 2x 1 + cos x - dx 59. If f is continuous and f(x) dx - 10, find Sof(2x) dx. 79. cot x dx 60. If f is continuous and J. f(x) dx - 4, find f3 xf(x?) dx. 81. fe dx x VIn x 61. If f is continuous on R, prove that 1 e+ dz [ ." f ( - x ) dx = [" f(x)dx Joetz For the case where f(x) = 0 and 0