-2 52. Given that sin*x dx = 8 7, what is sin*0 do? 0 4 53. In Example 5.1.2 we showed that fo x2 dx = 3. Use this fact and the properties of integrals to evaluate fo (5 - 6x2) dx. 54. Use the properties of integrals and the result of Example 4 to evaluate fi (2e* - 1) dx. 37-38 (a) Use the form of the definition of the integral given in Theo- 55. Use the result of Example 4 to evaluate J, e*+2 dx. rem 4 to evaluate the given integral. 56. Use the result of Exercise 47 and the fact that (b) Confirm your answer to part (a) graphically by interpreting the integral in terms of areas. Jo cos x dx = 1 (from Exercise 5.1.33), together with the properties of integrals, to evaluate " (2 cos x - 5x) dx. 37. [4x dx 38. * (2 - 2x) dx 57. Write as a single integral in the form S" f(x) dx: 39-40 [, f() ax + [i f()dx - [, f(x) dx (a) Find an approximation to the integral using a Riemann sum with right endpoints and n = 8. 58. If [2 f(x) dx = 7.3 and fi f(x) dx = 5.9, find [i f(x) dx. b) Draw a diagram like Figure 3 to illustrate the approximation in part (a). 59. If fof(x) dx = 37 and fog(x) dx = 16, find (c) Use Theorem 4 to evaluate the integral. (d) Interpret the integral in part (c) as a difference of areas and S. [2f (x) + 39 (x)] dx illustrate with a diagram like Figure 4. 60. Find fif(x) dx if 39. (3 - 2x) dx 40. [ (x2 - 3x) dx f (x ) = 1 3 for x 3 41-46 Evaluate the integral by interpreting it in terms of areas. 61. For the function f whose graph is shown, list the following 41. (10 - 5x) dx 42. (2x - 1) dx quantities in increasing order, from smallest to largest, and explain your reasoning. 43. 12x | dx 44. |2x - 1 / dx (A) Sof(x) dx (B) fif(x) dx (c) fif ( x) dx (D) [f (x) dx (E) f'(1) 45. (1 + 9 -x2) dx 46. ( 2x - V16 - x2 ) dx b 2 - a2 N 47. Prove that (x dx = 2 0 48. Prove that (x' dx = b' - a3 5 3