Questions 13, 15,16,18,25,26,29,30 384 Chapter Seven INTEGRATION 2+cos do 23. 22
ID: 2877659 • Letter: Q
Question
Questions 13, 15,16,18,25,26,29,30 384 Chapter Seven INTEGRATION 2+cos do 23. 22. 12. 2-sin 3 +sino dt 14. 72 du Estimate the values of the integrals in Exercises 26-27 decimal places by integrating the functions on your u+u 1 + 2 er compuer for lane tlues of th upper i 19. 27 e cos zdr dy 21. Problems 28. The graphs of y 1/a,y 1/2 and the functions For what values of p do the integrals in Problems 30-31 f(a), g(x), h(x), and k(r) are shown in Figure 7.25. verge or diverge? (a) Is the area between y = 1/z and 1-1/22 on the 1 to oc finite or infinite? Explain, (b) Using the graph, decide whether the integral of each of the functions f(z), g(z), h(a) and k(z) on the interval from 1 to oc converges, diverges, or dr dr interval from 30. a r(ln z) whether it is impossible to tell. 32. (a) Find an upper bound for - k(x) h(z) [Hint: e-/ e-3s for z 241 (b) For any positive n, generalize the result of part (a) find an upper bound for f(a) 1/r2 e dz Figure 7.25 by noting that nr s a for z 2 n. 33. In Planck's Radiation Law, we encounter the integra 29. Suppose ()dr converges. What does Figure 7.26 dz suggest about the convergence of J gx) dr? a) Explain why a graph of the tangent line toe at 9(r) tells us that for all t 70) (b) Substituting t 1/z, show that for all ne Figure 7.26 (c) Use the corExplanation / Answer
13)substitute t -5 =u =>dt = du
t =0 => u =0, t =8 => u =3
[5 to 8] 6/(t-5) dt
=[0 to 3] 6/u du
=[0 to 3] 6u-1/2 du
=[0 to 3] 6(1/(1+(-1/2)))u-(1/2)+1 +C
=[0 to 3] 6(1/(1/2))u1/2 +C
=[0 to 3] 6*2u1/2 +C
=[0 to 3] 12u1/2 +C
=12*31/2 +C -12*01/2 -C
=12*31/2
=123
[5 to 8] 6/(t-5) dt=123
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15)substitute t +1=u => dt =du
t =-1 => u =0, t =5 =>u =6
[-1 to 5] dt/(t+1)2
=[0 to 6] du/u2
=[0 to 6] u-2du
=limu->0+ [u to 6] (1/(-2+1))u-2+1+C
=limu->0+ [u to 6] -u-1+C
=limu->0+[-6-1+C]+[-u-1+C]
=-
integral diverges
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16)[- to ] du/(1+u2)
=[- to ] tan-1u +C
=tan-1 +C -tan-1(-) -C
=(/2)-(-/2)
=
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