41-46 Use the Comparison Theorem to determine whether the integral is convergent or divergent. 47. The integral 42. J 2 + ex x dx e arctan t 44. dx Jo 2+ e* sinx 46. dx 1 dx x (1 + x) is improper for two reasons: The interval [0, ) is infinite and the integrand has an infinite discontinuity at 0. Evalu- ate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: x (1 + x) dx = x ( 1 + x) 1 1 dx+ Si dx x (1 + x) 54. The average speed of molecules in an ideal gas is 4 M 3/2 2RT Sve-Mo/(2RT) dv where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and v is the molecular speed. Show that 56. Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of stars depends only on the distance r from the center of the cluster. If the perceived star density is given by y(s), where s is the observed planar distance from the center of the cluster, and x(r) is the actual density, it can be shown that 8RT TM 2r y(s) = [* x(r) dr If the actual density of stars in a cluster is x(r) = (R r), find the perceived density y(s). 58. Estimate the numerical value of fe dx by writing it as 60. Show that fe dx = In y dy by interpreting the the sum of edx and fedx. Approximate the first integral by using Simpson's Rule with n = 8 and show that -4x the second integral is smaller than ex dx, which is less than 0.0000001. integrals as areas. 62. Find the value of the constant C for which the integral x C dx +1 3x+1 converges. Evaluate the integral for this value of C. 64. Show that if a > -1 and b> a+ 1, then the following integral is convergent. xa dx Jo 1+x