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All Textbook Solutions for Calculus

Finding an Indefinite Integral In Exercises 1940, use a table of integrals to find the indefinite integral. x(x26x+10)2dx 34E 35E 36E 37E 38E 39E 40E 41E 42E Evaluating a Definite Integral In Exercises 4148, use a table of integrals to evaluate the definite integral. /2/2cosx1+sin2xdx 44E 45E 46E 47E 48E 49E 50E 51E Verifying a Formula In Exercises 49-54, verify the integration formula. (lnu)ndu=u(lnu)nn(lnu)n1du 53E 54E 55E 56E 57E 58E 59E 60E 61E 62E EXPLORING CONCEPTS Finding a Pattern (a) Find fxnlnxdx for n=1,2, and 3. Describe any patterns you notice. (b) Write a general rule for evaluating the integral in part (a) for an integer n1. (c) Verify your rule from part (b) using integration by parts.64E 65E 66E 67E 68E 69E 70E 73E 71E Building Design The cross section of a precast concrete beam for a building is bounded by the graphs of the equations x=21+y2,x=21+y2,y=0,andy=3 where x and y are measured in feet. The length of the beam is 20 feet (see figure). (a) Find the volume V and the weight W of the beam. Assume the concrete weighs 148 pounds per cubic foot. (b) Find the centroid of a cross section of the beam.74E Determining Whether an Integral Is Improper In Exercises 512, decide whether the integral is improper. Explain your reasoning. 01dx5x3 2E 3E Determining Whether an Integral Is Improper In Exercises 512, decide whether the integral is improper. Explain your reasoning. 1lnx2dx 5E 6E Determining Whether an Integral Is Improper In Exercises 512, decide whether the integral is improper. Explain your reasoning. sinx4+x2dx 8E 9E Evaluating an Improper Integral In Exercises 13-16, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 341(x3)3/2dx Evaluating an Improper Integral In Exercises 13-16, explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. 021(x1)2dx 12E 13E 14E Writing In Exercises 1316, explain why the evaluation of the integral is incorrect. Use the integration capabilities of a graphing utility to attempt to evaluate the integral. Determine whether the utility gives the correct answer.16E 17E 18E 19E 20E 21E 22E Evaluating an Improper Integral In Exercises 1732, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 0x2exdx 24E Evaluating an Improper Integral In Exercises 1732, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 41x(lnx)3dx Evaluating an Improper Integral In Exercises 1732, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. 1lnxxdx 27E 28E 29E 30E 31E 32E 33E 34E 35E Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0838xdx Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 01xlnxdx Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0elnx2dx Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 0/2tand 40E 41E 42E Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 351x29dx Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 05125x2dx Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 31xx29dx 46E Evaluating an Improper Integral In Exercises 3348, determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. 04x(x+6)dx 48E Finding Values In Exercises 49 and 50, determine all values of p for which the improper integral converges. 11xpdx 50E 51E 52E 53E Convergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 011x9dx 55E Convergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 0x4exdx 57E 58E 59E Convergence or Divergence In Exercises 53–62, use the results of Exercises 49–52 to determine whether the improper integral converges or diverges. 60. Convergence or Divergence In Exercises 5360, use the results of Exercises 4952 to determine whether the improper integral converges or diverges. 11sinxx2dx 62E 63E 64E 65E 66E Area In Exercises 6770, find the area of the unbounded shaded region. y=ex,x1 68E Area In Exercises 63-66, find the area of the unbounded shaded region. Witch of Agnesi:Area In Exercises 63-66, find the area of the unbounded shaded region. Witch of Agensi: y=8x2+4 Area and Volume In Exercises 67 and 68, consider the region satisfying the inequalities (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the x -axis. (c) Find the volume of the solid generated by revolving the region about the y -axis. yex,y0,x0 72E Arc Length Sketch the graph of the hypocycloid of four cusps x2/3 + y2/3 = 4 and find its perimeter.74E 75E 76E 77E Propulsion In Exercises 77 and 78, use the weight of the rocket to answer each question. (Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earths surface? (b) How far has the rocket traveled when half the total work has occurred? 10-ton rocket 79E 80E Capitalized Cost In Exercises 81 and 82, find the capitalized cost C of an asset (a) for n = 5 years, (b) for n = 10 years, and (c) forever. The capitalized cost is given by c=c0+n0c(t)ertdt where C0 is the original investment, t is the time in years, r is the annual interest rate compounded continuously, and c(t) is the annual cost of maintenance. c0=650,000c(t)=25,000r=0.06 Capitalized Cost In Exercises 81 and 82, find the capitalized cost C of an asset (a) for n = 5 years, (b) for n = 10 years, and (c) forever. The capitalized cost is given by C=C0+n0c(t)ertdt where C0 is the original investment, t is the time in years, r is the annual interest rate compounded continuously, and c(t) is the annual cost of maintenance. C0=650,000c(t)=25,000(1+0.08t)r=0.06 83E 84E 85E 86E 87E 88E 89E Making an Integral Improper For each integral, find a nonnegative real number b that makes the integral improper. Explain your reasoning. b01x29dx b014xdx b0xx27x+12dx 10blnxdx b0tan2xdx b0cosx1sinxdx 91E 92E 93E 94E 95E 96E 97E 98E 99E 100E 101E 102E 103E 104E 105E 106E 107E 108E u -Substitution In Exercises 105 and 106, rewrite the improper integral as a proper integral using the given u -substitution. Then use the Trapezoidal Rule with n = 5 to approximate the integral. 01sinxxdx,u=x 110E 111E 1RE 2RE 3RE 4RE Using Basic Integration Rules In Exercises 18, use the basic integration rules to find or evaluate the integral. 1eln2xxdx 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE Using Partial Fractions In Exercises 3744, use partial fractions to find the indefinite integral. x2+2xx3x2+x1dx 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE Verifying a Formula Verify the reduction formula (lnx)ndx=x(lnx)nn(lnx)n1dx.48RE 49RE 50RE 51RE 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE 64RE 65RE 66RE 67RE 68RE 69RE 70RE 71RE 72RE 73RE 74RE 75RE 76RE 77RE 78RE 79RE 80RE 81RE 82RE 83RE 84RE 85RE 86RE 87RE 88RE Present Value The board of directors of a corporation is calculating the price to pay for a business that is forecast to yield a continuous flow of profit of $500,000 per year. The money will earn a nominal rate of 5% per year compounded continuously. The present value of the business for t0 years is Present value =0t0500,000e0.05tdt. (a) Find the present value of the business for 20 years. (b) Find the present value of the business in perpetuity (forever).90RE 91RE 1PS 2PS 3PS 4PS 5PS 6PS Area Consider the problem of finding the area of the region bounded by the x-axis, the line x = 4, and the curve y=x2(x2+9)3/2.Area Use the substitution u=tanx2 v to find the area of the shaded region under the graph of y=12+cosx for 0x/2 (see figure).9PS 10PS 11PS 12PS 13PS 14PS 15PS 16PS 17PS 18PS 19PS 20PS 21PS 22PS Writing the Terms of a Sequence In Exercises 510, write the first five terms of the sequence with the given n th term. an=2n 2E Writing the Terms of a Sequence In Exercises 510, write the first five terms of the sequence with the given n th term. an=sinn2 4E 5E 6E Writing the Terms of a Sequence In Exercises 11 and 12, write the first five terms of the recursively defined sequence. a1=3,ak1=2(a11)8E 9E Matching In Exercises 13-16, match the sequence with the given nth term with its graph. [The graphs are labeled (a), (b), (c). and (d).] an=10nn+1 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E Finding the Limit of a Sequence In Exercises 2124, find the limit of the sequence with the given n th term. an=cos2n 25E 26E 27E 28E Determining Convergence or Divergence In Exercises 2944, determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit. an=5n+2 30E 31E 32E 33E Determining Convergence or Divergence In Exercises 2944, determine the convergence or divergence of the sequence with the given n th term. If the sequence converges, find its limit. an=n3n3+1

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