41-46 Use the Comparison Theorem to determine whether the integral is convergent or divergent. 47. The integral dx 42. 2 + ex x x (1 + x) dx 44. e arctan x dx Jo 2+ ex 46. for sinx dx Jo 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: So x ( 1 + x) dx= dx + x (1 + x) S x Q + x d dx 54. The average speed of molecules in an ideal gas is 4 M 3/2 v ve-Mo/(2RT) dv 2RT 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 v 8RT 2 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). integrals as areas. 58. Estimate the numerical value of forex dx by writing it as 60. Show that fo ex 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 the second integral is smaller than ex dx, which is less than 0.0000001. 62. Find the value of the constant C for which the integral (+1) 3x+1 dx 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 +xb dx