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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Surface area Let f(x)=x+1. Find the area of the surface generated when the region bounded by the graph of f on the interval [0, 1] is revolved about the x-axis.Surface area Find the area of the surface generated when the region bounded by the graph of y=ex+14ex on the interval [0, ln 2] is revolved about the x-axis.Arc length Find the length of the curve y = x5/4 on the interval [0, 1]. (Hint: Write the arc length integral and let u2=1+(54)2x.)Skydiving A skydiver in free fall subject to gravitational acceleration and air resistance has a velocity given by v(t)=vT(eat1eat+1), where vT is the terminal velocity and a 0 is a physical constant. Find the distance that the skydiver falls after t seconds, which is d(t)=0tv(y)dy.What are the best choices for u and dv in evaluating xcosxdx?Verify by differentiation that lnxdx=xlnxx+C.How many times do you need to integrate by parts to reduce 1e(lnx)6dx to an integral of ln x?On which derivative rule is integration by parts based?Use integration by parts to evaluate xcosxdx with u = x and dv = cos x dx.Use integration by parts to evaluate xlnxdx with u = ln x and dv = x dx.Explain how integration by parts is used to evaluate a definite integral.5EHow would you choose dv when evaluating xneaxdx using integration by parts?Integrals involving lnxdx Use a substitution to reduce the following integrals to lnudu. Then evaluate the resulting integral. 53. (sec2x)ln(tanx+2)dxIntegrals involving lnxdx Use a substitution to reduce the following integrals to lnudu. Then evaluate the resulting integral. 52. (cosx)ln(sinx)dxIntegration by partsEvaluate the following integrals using integration by parts. 9. xcos5xdxIntegration by parts Evaluate the following integrals. 8. xsin2xdxIntegration by partsEvaluate the following integrals using integration by parts. 11. te6tdtIntegration by parts Evaluate the following integrals. 10. 2xe3xdxIntegration by partsEvaluate the following integrals using integration by parts. 13. xln10xdxIntegration by parts Evaluate the following integrals. 12. se2sdsIntegration by partsEvaluate the following integrals using integration by parts. 15. (2w+4)cos2wdwIntegration by parts Evaluate the following integrals. 14. sec2dIntegration by partsEvaluate the following integrals using integration by parts. 17. x3xdxIntegration by partsEvaluate the following integrals using integration by parts. 18. x9lnxdxIntegration by parts Evaluate the following integrals. 17. lnxx10dxIntegration by parts Evaluate the following integrals. 18. sin1xdxIntegration by parts Evaluate the following integrals. 21. xsinxcosxdxIntegration by partsEvaluate the following integrals using integration by parts. 22. e2xsinexdxRepeated integration by parts Evaluate the following integrals. 29. x2sin2xdxRepeated integration by parts Evaluate the following integrals. 30. x2e4xdxRepeated integration by parts Evaluate the following integrals. 23. t2etdtIntegration by partsEvaluate the following integrals using integration by parts. 26. t3sintdtIntegration by partsEvaluate the following integrals using integration by parts. 27. excosxdxRepeated integration by parts Evaluate the following integrals. 24. e3xcos2xdxRepeated integration by parts Evaluate the following integrals. 25. exsin4xdxRepeated integration by parts Evaluate the following integrals. 28. e20sin6dIntegration by partsEvaluate the following integrals using integration by parts. 31. e3xsinexdxIntegration by partsEvaluate the following integrals using integration by parts. 32. 01x22xdxDefinite integrals Evaluate the following definite integrals. 31. 0xsinxdxDefinite integrals Evaluate the following definite integrals. 32. 1eln2xdxDefinite integrals Evaluate the following definite integrals. 33. 0/2xcos2xdxDefinite integrals Evaluate the following definite integrals. 34. 0ln2xexdxDefinite integrals Evaluate the following definite integrals. 35. 1e2x2lnxdxRepeated integration by parts Evaluate the following integrals. 26. x2ln2xdxIntegration by partsEvaluate the following integrals using integration by parts. 39. 01sin1ydyIntegration by partsEvaluate the following integrals using integration by parts. 40. exdxEvaluate the integral in part (a) and then use this result to evaluate the integral in part (b). a. tan1xdx b. xtan1x2dxVolumes of solidsFind the volume of the solid that is generated when the given region is revolved as described. 42. The region bounded by f(x) = ln x, y = 1, and the coordinate axes is revolved about the x-axis.Volumes of solids Find the volume of the solid that is generated when the given region is revolved as described. 39. The region bounded by f(x) = ex, x = ln 2, and the coordinate axes is revolved about the y-axis.Volumes of solids Find the volume of the solid that is generated when the given region is revolved as described. 40. The region bounded by f(x) = sin x and the x-axis on [0, ] is revolved about the y-axis.Volumes of solidsFind the volume of the solid that is generated when the given region is revolved as described. 45. The region bounded by g(x)=lnx and the x-axis on [1, e] is revolved about the x-axis.Volumes of solids Find the volume of the solid that is generated when the given region is revolved as described. 42. The region bounded by f(x) = ex and the x-axis on [0, ln 2] is revolved about the line x = ln 2.Volumes of solids Find the volume of the solid that is generated when the given region is revolved as described. 41. The region bounded by f(x) = x ln x and the x-axis on [1, e2] is revolved about the x-axis.Integral of sec3 x Use integration by parts to show that sec3xdx=12secxtanx+12secxdx.Reduction formulas Use integration by parts to derive the following reduction formulas. 44. xneaxdx=xneaxanaxn1eaxdx, for a 0Reduction formulas Use integration by parts to derive the following reduction formulas. 45. xncosaxdx=xnsinaxanaxn1sinaxdx, for a 0Reduction formulas Use integration by parts to derive the following reduction formulas. 46. xnsinaxdx=xncosaxa+naxn1cosaxdx, for a 0Reduction formulas Use integration by parts to derive the following reduction formulas. 47. lnnxdx=xlnnxnlnn1xdxApplying reduction formulas Use the reduction formulas in Exercises 50-53 to evaluate the following integrals. 54. x2e3xdxApplying reduction formulas Use the reduction formulas in Exercises 50-53 to evaluate the following integrals. 55. x2cos5xdxApplying reduction formulas Use the reduction formulas in Exercises 50-53 to evaluate the following integrals. 56. x3sinxdxApplying reduction formulas Use the reduction formulas in Exercises 50-53 to evaluate the following integrals. 57. 1eln3xdxTwo methods Evaluate 0/3sinxln(cosx)dx in the following two ways. a. Use integration by parts. b. Use substitution.Two methods a. Evaluate xx+1dx using integration by parts. b. Evaluate xx+1dx using substitution. c. Verify that your answers to part (a) and (b) are consistent.Two methods a. Evaluate xlnx2dx using the substitution u = x2 and evaluating lnudu. b. Evaluate xlnx2dx using integration by parts. c. Verify that your answers to parts (a) and (b) are consistent.Logarithm base b Prove that logbxdx=1lnb(xlnxx)+C.Two integration methods Evaluate sinxcosxdx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers.Combining two integration methods Evaluate cosxdx using a substitution followed by integration by parts.64EAn identity Show that if f has a continuous second derivative on [a, b] and f(a) = f(b) = 0, then abxf(x)dx=f(a)f(b).Integrating derivatives Use integration by parts to show that if f is continuous on [a, b], then abf(x)f(x)dx=12(f(b)2f(a)2).Function defined as an integral Find the arc length of the function f(x)=exln2t1dt on [e, e3].Log integrals Use integration by parts to show that for m 1, xmlnxdx=xm+1m+1(lnx1m+1)+C and for m = 1, lnxxdx=12ln2x+C.Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, ]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis?A useful integral a. Use integration by parts to show that if f is continuous, xf(x)dx=xf(x)f(x)dx. b. Use part (a) to evaluate xe3xdx.Solid of revolution Find the volume of the solid generated when the region bounded by y = cos x and the x-axis on the interval [0, /2] is revolved about the y-axis.72ETwo useful exponential integrals Use integration by parts to derive the following formulas for real numbers a and b. eaxsinbxdx=eax(asinbxbcosbx)a2+b2+Ceaxcosbxdx=eax(acosbx+bsinbx)a2+b2+CIntegrating inverse functions Assume that f has an inverse on its domain. a. Let y = f1(x) and show that f1(x)dx=yf(y)dy. b. Use part (a) to show that f1(x)dx=yf(y)f(y)dy. c. Use the result of part (b) to evaluate lnxdx (express the result in terms of x). d. Use the result of part (b) to evaluate sin1xdx. e. Use the result of part (b) to evaluate tan1xdx.75EFind the error Suppose you evaluate dxx using integration by parts. With u = 1/x and dv = dx, you find that du = 1/x2 dx, v = x, and dxx=(1x)xx(1x2)dx=1+dxx. You conclude that = 1. Explain the problem with the calculation.77EPractice with tabular integration Evaluate the following integrals using tabular integration (refer to Exercise 77). a. x4exdx b. 7xe3xdx c. 102x2x+1dx d. (x32x)sin2xdx e. 2x23x(x1)3 f. x2+3x+432x+1dx g. Why doesnt tabular integration work well when applied to x1x2dx? Evaluate this integral using a different method. Tabular integration Consider the integral f(x)g(x)dx, where f can be differentiated repeatedly and g can be integrated repeatedly. Let Gk represent the result of calculating k Indefinite integrals of g, where the constants of integration are omitted. a. Show that integration by parts, when applied to f(x)g(x)dx with the choices u = f(x) and dv = g(x) dx, leads to f(x)g(x)dx = f(x)G1(x) f(x)G1(x)dx. This formula can be remembered by utilizing the following table, where a right arrow represents a product of functions on the right side of the integration by pats formula and a left arrow represents the integral of a product of functions (also appearing on the right side of the formula). Explain the significance of the signs associated with the arrows. b. Perform integration by parts again on f(x)G1(x)dx (from part (a)) with the choices u = f(x) and dv = G1(x) dx to Show that f(x)g(x)dx= f(x)G1(x) f (x)G2(x) + f(x)G2(x)dx. Explain the connection between this integral formula and the following table, paying close attention to the signs attached to the arrows. c. Continue the pattern established in parts (a) and (b) and integrate by parts a third time. Write the integral formula that results from three applications of integration by parts, and construct the associated tabular integration table (include signs of the arrows) d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts. Evaluate x2ex/2dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in e. Use tabular integration to evaluate x3cosxdx. How many rows of the table are necessary? Why? f. Explain why tabular integration is particularly suited to integrals of the form pn(x)g(x)dx, where pn is a polynomial of degree n 0 (and where, as before, we assume g is easily integrated as many times as necessary)Tabular integration extended Refer to Exercise 77. a. The following table shows the method of tabular integration applied to excosxdx. Use the table to express excosxdxin terms of the sum of functions and an indefinite integral. b. Solve the equation in part (a) for excosxdx. c. Evaluate e2xsin3xdx by applying the idea from parts (a) and (b). Tabular integration Consider the integral f(x)g(x)dx, where f can be differentiated repeatedly and g can be integrated repeatedly. Let Gk represent the result of calculating k Indefinite integrals of g, where the constants of integration are omitted. a. Show that integration by parts, when applied to f(x)g(x)dx with the choices u = f(x) and dv = g(x) dx, leads to f(x)g(x)dx = f(x)G1(x) f(x)G1(x)dx. This formula can be remembered by utilizing the following table, where a right arrow represents a product of functions on the right side of the integration by pats formula and a left arrow represents the integral of a product of functions (also appearing on the right side of the formula). Explain the significance of the signs associated with the arrows. b. Perform integration by parts again on f(x)G1(x)dx (from part (a)) with the choices u = f(x) and dv = G1(x) dx to Show that f(x)g(x)dx= f(x)G1(x) f(x)G2(x) + f(x)G2(x)dx. Explain the connection between this integral formula and the following table, paying close attention to the signs attached to the arrows. c. Continue the pattern established in parts (a) and (b) and integrate by parts a third time. Write the integral formula that results from three applications of integration by parts, and construct the associated tabular integration table (include signs of the arrows) d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts. Evaluate x2ex/2dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in e. Use tabular integration to evaluate x3cosxdx. How many rows of the table are necessary? Why? f. Explain why tabular integration is particularly suited to integrals of the form pn(x)g(x)dx, where pn is a polynomial of degree n 0 (and where, as before, we assume g is easily integrated as many times as necessary)An identity Show that if f and g have continuous second derivatives and f(0) = f(1) = g(0) = g(1) = 0, then 01f(x)g(x)dx=01f(x)g(x)dx.Possible and impossible integrals Let In=xnex2dx, where n is a nonnegative integer. a. I0=ex2dx cannot be expressed in terms of elementary functions. Evaluate I1. b. Use integration by parts to evaluate I3. c. Use integration by parts and the result of part (b) to evaluate I5. d. Show that, in general, if n is odd, then In=12ex2pn1(x), where pn1 is a polynomial of degree n 1. e. Argue that if n is even, then In cannot be expressed in terms of elementary functions.A family of exponentials The curves y = xeax are shown in the figure for a = 1, 2, and 3. a. Find the area of the region bounded by y = xex and the x-axis on the interval [0, 4]. b. Find the area of the region bounded by y = xeax and the x-axis on the interval [0, 4], where a 0. c. Find the area of the region bounded by y = xeax and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b), where a 0 and b 0. d. Use part (c) to show that A(l, ln b) = 4A(2, (ln b)/2). e. Does this pattern continue? Is it true that A(1, ln b) = a2A(a, (ln b)/a)?Evaluate sin3xdxby splitting off a factor of sin x rewriting sin2x in terms of cos x and using an appropriate u-substitution.What strategy would you use to evaluate sin3xcos3xdx?State the half-angle identities used to integrate sin2 x and cos2 x.State the three Pythagorean identities.Describe the method used to integrate sin3 x.Describe the method used to integrate sinm x cosn x, for m even and n odd.What is a reduction formula?How would you evaluate cos2xsin3xdx?How would you evaluate tan10xsec2xdx?How would you evaluate sec12xtanxdx?Integrals of sin x or cos x Evaluate the following integrals. 11. cos3xdxIntegrals of sin x or cos x Evaluate the following integrals. 10. sin3xdxTrigonometric integralsEvaluate the following integrals. 11. sin2x3xdxIntegrals of sin x or cos x Evaluate the following integrals. 12. cos42dIntegrals of sin x or cos x Evaluate the following integrals. 13. sin5xdxIntegrals of sin x or cos x Evaluate the following integrals. 14. cos320xdxIntegrals of sin x and cos x Evaluate the following integrals. 17. sin3xcos2xdxIntegrals of sin x and cos x Evaluate the following integrals. 18. sin2cos5dIntegrals of sin x and cos x Evaluate the following integrals. 19. cos3xsinxdxIntegrals of sin x and cos x Evaluate the following integrals. 20. sin3cos2dTrigonometric integrals Evaluate the following integrals. 19. 0/3sin5xcos-2xdxIntegrals of sin x and cos x Evaluate the following integrals. 22. sin3/2xcos3xdxTrigonometric integrals Evaluate the following integrals. 21. 0/2cos3xsin3xdx22EIntegrals of sin x and cos x Evaluate the following integrals. 15. sin2xcos2xdxIntegrals of sin x and cos x Evaluate the following integrals. 16. sin3xcos5xdxIntegrals of sin x and cos x Evaluate the following integrals. 23. sin2xcos4xdxIntegrals of sin x and cos x Evaluate the following integrals. 24. sin3xcos3/2xdxIntegrals of tan x or cot x Evaluate the following integrals. 25. tan2xdxIntegrals of tan x or cot x Evaluate the following integrals. 26. 6sec4xdxIntegrals of tan x or cot x Evaluate the following integrals. 27. cot4xdxIntegrals of tan x or cot x Evaluate the following integrals. 28. tan3dIntegrals of tan x or cot x Evaluate the following integrals. 29. 20tan6xdxIntegrals of tan x or cot x Evaluate the following integrals. 30. cot53xdxIntegrals involving tan x and sec x Evaluate the following integrals. 31. 10tan9xsec2xdxIntegrals involving tan x and sec x Evaluate the following integrals. 32. tan9xsec4xdxIntegrals involving tan x and sec x Evaluate the following integrals. 33. tanxsec3xdxTrigonometric integrals Evaluate the following integrals. 36. tan4xsec3/24xdxAdditional integrals Evaluate the following integrals. 51. sec4(ln)dIntegrals involving tan x and sec x Evaluate the following integrals. 42. tan5sec4dAdditional integrals Evaluate the following integrals. 53. /3/3sec21dTrigonometric integrals Evaluate the following integrals. 40. 0/6tan52xsec2xdxTrigonometric integrals Evaluate the following integrals. 41. 0/4sec7xsecxdxIntegrals involving tan x and sec x Evaluate the following integrals. 34. tanxsec4xdxIntegrals involving tan x and sec x Evaluate the following integrals. 35. tan34xdxIntegrals involving tan x and sec x Evaluate the following integrals. 36. sec2xtan5xdxIntegrals involving tan x and sec x Evaluate the following integrals. 37. sec2xtan1/2xdxIntegrals involving tan x and sec x Evaluate the following integrals. 38. sec2xtan3xdxIntegrals involving tan x and sec x Evaluate the following integrals. 39. csc4xcot2xdxIntegrals involving tan x and sec x Evaluate the following integrals. 40. csc10xcotxdxTrigonometric integrals Evaluate the following integrals. 49. /20/10csc25wcot45wdwTrigonometric integrals Evaluate the following integrals. 50. csc10xcot3xdxTrigonometric integrals Evaluate the following integrals. 51. (csc2x+csc4x)dxTrigonometric integrals Evaluate the following integrals. 52. 0/8(tan2x+tan32x)dxIntegrals involving tan x and sec x Evaluate the following integrals. 41. 0/4sec4dAdditional integrals Evaluate the following integrals. 50. 0/2xsin3(x2)dxIntegrals involving tan x and sec x Evaluate the following integrals. 43. /6/3cot3dIntegrals involving tan x and sec x Evaluate the following integrals. 44. 0/4tan3sec2dAdditional integrals Evaluate the following integrals. 55. 0(1cos2x)3/2dx58ESquare roots Evaluate the following integrals. 59. 0/21cos2xdxSquare roots Evaluate the following integrals. 60. 0/81cos8xdxSquare roots Evaluate the following integrals. 61. 0/4(1+cos4x)3/2dxArc length Find the length of the curve y = ln (sec x), for 0 x /4.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If m is a positive integer, then 0cos2m+1xdx=0. b. If m is a positive integer, then 0sinmxdx=0.Sine football Find the volume of the solid generated when the region bounded by the graph of y = sin x and the x-axis on the interval [0, ] is revolved about the x-axis.VolumeFind the volume of the solid generated when the region bounded by y = sin2 x cos 3/2 x and the x-axis on the interval [0, /2] is revolved about the x-axis.66EIntegrals of the form sinmxcosnxdx Use the following three identities to evaluate the given integrals. sinmxsinnx=12(cos((mn)x)cos((m+n)x))sinmxcosnx=12(sin((mn)x)+sin((m+n)x))cosmxcosnx=12(cos((mn)x)+cos((m+n)x)) 67. sin3xcos7xdxIntegrals of the form sinmxcosnxdx Use the following three identities to evaluate the given integrals. sinmxsinnx=12(cos((mn)x)cos((m+n)x))sinmxcosnx=12(sin((mn)x)+sin((m+n)x))cosmxcosnx=12(cos((mn)x)+cos((m+n)x)) 68. sin5xsin7xdxIntegrals of the form sinmxcosnxdx Use the following three identities to evaluate the given integrals. sinmxsinnx=12(cos((mn)x)cos((m+n)x))sinmxcosnx=12(sin((mn)x)+sin((m+n)x))cosmxcosnx=12(cos((mn)x)+cos((m+n)x)) 69. sin3xsin2xdxIntegrals of the form sinmxcosnxdx Use the following three identities to evaluate the given integrals. sinmxsinnx=12(cos((mn)x)cos((m+n)x))sinmxcosnx=12(sin((mn)x)+sin((m+n)x))cosmxcosnx=12(cos((mn)x)+cos((m+n)x)) 70. cosxcos2xdxIntegrals of the form sinmxcosnxdx Use the following three identities to evaluate the given integrals. sinmxsinnx=12(cos((mn)x)cos((m+n)x))sinmxcosnx=12(sin((mn)x)+sin((m+n)x))cosmxcosnx=12(cos((mn)x)+cos((m+n)x)) 71. Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m n. a. 0sinmxsinnxdx=0 b. 0cosmxcosnxdx=0 c. 0sinmxcosnxdx=0, for |m + n| even72EA tangent reduction formula Prove that for positive integers n 1, tannxdx=tann1xn1tann2xdx. Use the formula to evaluate 0/4tan3xdx.A secant reduction formula Prove that for positive integers n 1, secnxdx=secn2xtanxn1+n2n1secn2xdx. (Hint: Integrate by parts with u = secn 2 x and dv = sec2 x dx.)75EUse a substitution of the form x = a sin to transform 9 x2 into a product.2QCThe integral dxa2+x21atan1xa+C is given in Section 5.5. Verify this result with the appropriate trigonometric substitution.What change of variables is suggested by an integral containing x29?What change of variables is suggested by an integral containing x2+36?What change of variables is suggested by an integral containing 100x2?If x = 4 tan , express sin in terms of x.If x = 2 sin , express cot in terms of x.If x = 8 sec , express tan in terms of x.Sine substitution Evaluate the following integrals. 7. 05/2dx25x2Sine substitution Evaluate the following integrals. 8. 03/2dx(9x2)3/2Sine substitution Evaluate the following integrals. 9. 510100x2dxSine substitution Evaluate the following integrals. 10. 02x24x2dxSine substitution Evaluate the following integrals. 11. 01/2x21x2dxSine substitution Evaluate the following integrals. 12. 1/211x2x2dxSine substitution Evaluate the following integrals. 13. dx(16x2)1/2Sine substitution Evaluate the following integrals. 14. 36t2dtTrigonometric substitutions Evaluate the following integrals. 21. dxx2x2+9Trigonometric substitutions Evaluate the following integrals. 39. x2(25+x2)2dxTrigonometric substitutions Evaluate the following integrals using trigonometric substitution. 17. 02x2x2+4dxTrigonometric substitutions Evaluate the following integrals. 20. dx(1+x2)3/2Trigonometric substitutions Evaluate the following integrals. 25. dxx281,x9Trigonometric substitutions Evaluate the following integrals. 18. dxx249,x7Trigonometric substitutions Evaluate the following integrals. 17. 64x2dxTrigonometric substitutions Evaluate the following integrals. 22. dtt29t2Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. 23. dx(25x2)3/2Trigonometric substitutions Evaluate the following integrals. 38. 9x2x2dxTrigonometric substitutions Evaluate the following integrals using trigonometric substitution. 25. 9x2xdxEvaluating definite integrals Evaluate the following definite integrals. 52. 22x21xdxTrigonometric substitutions Evaluate the following integrals using trigonometric substitution. 27. 01/3dx(9x2+1)3/2Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. 28. 06z2(z2+36)2dzTrigonometric substitutions Evaluate the following integrals using trigonometric substitution. 29. dx(4+x2)2Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. 30. x31x2dxTrigonometric substitutions Evaluate the following integrals. 29. x216x2dxTrigonometric substitutions Evaluate the following integrals. 28. dx(x236)3/2,x6Trigonometric substitutions Evaluate the following integrals. 31. x29xdx,x3Trigonometric substitutions Evaluate the following integrals. 44. dxx3x21,x1Trigonometric substitutions Evaluate the following integrals. 45. dxx(x21)3/2,x1Evaluating definite integrals Evaluate the following definite integrals. 48. 8216dxx264Evaluating definite integrals Evaluate the following definite integrals. 49. 1/31dxx21x2Evaluating definite integrals Evaluate the following definite integrals. 50. 12dxx24x2Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. 39. x2(100x2)3/2dxEvaluating definite integrals Evaluate the following definite integrals. 54. 10/310dyy225Trigonometric substitutions Evaluate the following integrals. 27. dx(1+4x2)3/2Trigonometric substitutions Evaluate the following integrals. 40. dxx29x21,x13Trigonometric substitutions Evaluate the following integrals using trigonometric substitution. 43. 04/3dxx2+16Trigonometric substitutions Evaluate the following integrals. 24. dx16+4x2Trigonometric substitutions Evaluate the following integrals. 43. x3(81x2)2dx46EEvaluating definite integrals Evaluate the following definite integrals. 55. 4/34dxx2(x24)Trigonometric substitutions Evaluate the following integrals. 32. 94x2dxEvaluating definite integrals Evaluate the following definite integrals. 51. 01/3x2+1dxSine substitution Evaluate the following integrals. 16. (369x2)3/2dxTrigonometric substitutions Evaluate the following integrals. 33. x24+x2dx52ETrigonometric substitutions Evaluate the following integrals. 37. 9x225x3dx,x5354ETrigonometric substitutions Evaluate the following integrals. 42. dxx3x2100,x1056EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If x = 4 tan , then csc = 4/x. b. The integral 121x2dx does not have a finite real value. c. The integral 12x21dx does not have a finite real value. d. The integral dxx2+4x+9 cannot be evaluated using trigonometric substitution.Area of an ellipse The upper half of the ellipse centered at the origin with axes of length 2a and 2b is described by y=baa2x2 (see figure). Find the area of the ellipse in terms of a and b.Area of a segment of a circle Use two approaches to show that the area of a cap (or segment) of a circle of radius r subtended by an angle (see figure) is given by Aseg=12r2(sin). a. Find the area using geometry (no calculus). b. Find the area using calculus.Completing the square Evaluate the following integrals. 58. dxx26x+34Completing the squareEvaluate the following integrals. 61. dx32xx2Completing the square Evaluate the following integrals. 60. du2u212u+36Completing the square Evaluate the following integrals. 59. dxx2+6x+18Completing the square Evaluate the following integrals. 61. x22x+1x22x+10dxCompleting the square Evaluate the following integrals. 65. 1/2(2+3)/(22)dx8x28x+11Completing the square Evaluate the following integrals. 64. 14dtt22t+10Completing the square Evaluate the following integrals. 63. x28x+16(9+8xx2)3/2dxAsymmetric integrands Evaluate the following integrals. Consider completing the square. 76. dx(x1)(3x)Asymmetric integrands Evaluate the following integrals. Consider completing the square. 77. 2+24dx(x1)(x3)Using the integral of sec3 u By reduction formula 4 in Section 7.3, sec3udu=12(secutanu+lnsecu+tanu)+C. Graph the following functions and find the area under the curve on the given interval. 74. f(x) = (4 + x2)1/2, [0, 2]Using the integral of sec3 u By reduction formula 4 in Section 7.3, sec3udu=12(secutanu+lnsecu+tanu)+C. Graph the following functions and find the area under the curve on the given interval. 73. f(x)=(9x2)2,[0,32]72E73EUsing the integral of sec3 uBy reduction formula 4 in Section 8.3, sec3udu=12(secutanu+ln|secu+tanu|)+C. Graph the following functions and find the area under the curve on the given interval. 74. f(x)=1xx236, [123,12]75EArea and volume Consider the function f(x) = (9 + x2)1/2 and the region R on the interval [0, 4] (see figure). a. Find the area of R. b. Find the volume of the solid generated when R is revolved about the x-axis. c. Find the volume of the solid generated when R is revolved about the y-axis.Arc length of a parabola Find the length of the curve y = ax2 from x = 0 to x = 10, where a 0 is a real number.78EShow that dxxx21={sec1x+C=tan1x21+Cifx1sec1x+C=tan1x21+Cifx1.Evaluate for x21x3dx, for x 1 and for x 1.81EMagnetic field due to current in a straight wire A long, straight wire of length 2L on the y-axis carries a current L. According to the Biot-Savart Law, the magnitude of the magnetic field due to the current at a point (a, 0) is given by B(a)=0IrLLsinr2dy, where 0 is a physical constant, a 0, and , r, and y are related as shown in the figure. a. Show that the magnitude of the magnetic field at (a, 0) is B(a)=0IL2aa2+L2. b. What is the magnitude of the magnetic field at (a, 0) due to an infinitely long wire (L )?83E85E86EFind an antiderivative of f(x)=1x2+2x+4.If the denominator of a reduced proper rational function is (x 1) (x + 5)(x 10), what is the general form of its partial fraction decomposition?3QC4QCWhat kinds of functions can be integrated using partial fraction decomposition?Give an example of each of the following. a. A simple linear factor b. A repeated linear factor c. A simple irreducible quadratic factor d. A repeated irreducible quadratic factorWhat term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following? a. A factor of x 3 in the denominator b. A factor of (x 4)3 in the denominator c. A factor of x2 + 2x + 6 in the denominator4ESet up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 5. 4xx29x+20Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 6. 4x+14x21Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 7. x+3(x5)2Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 8. 2x32x2+xSet up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 9. 4x55x3+4xSetting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the folio wing functions. 39. 20x(x1)2(x21)Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 11. 1x(x2+1)Setting up partial fraction decompositions Give the appropriate form of the partial fraction decomposition for the folio wing functions. 41. 2x2+3(x28x+16)(x2+3x+4)Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 13. x4+12x2(x2)2(x2+x+2)214ESet up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 15. x(x416)2Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants. 16. x2+2x+6x3(x2+x+1)2Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 7. 5x7x23x+2Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 8. 11x10x2xGive the partial fraction decomposition for the following expressions. 19. 6x22x8Setting up partial fraction decomposition Give the partial fraction decomposition for the following functions. 12. x24x+11(x3)(x1)(x+1)Give the partial fraction decomposition for the following expressions. 21. 2x2+5x+6x2+3x+2 (Hint: Use long division first.)Give the partial fraction decomposition for the following expressions. 22. x4+2x3+xx21IntegrationEvaluate the following integrals. 23.3(x1)(x+2)dxIntegrationEvaluate the following integrals. 24. 8(x2)(x+6)dxIntegrationEvaluate the following integrals. 25. 6x21dxSimple linear factors Evaluate the following integrals. 16. 01dtt29IntegrationEvaluate the following integrals. 27. 8x53x25x+2dxIntegrationEvaluate the following integrals. 28. 127x23x22xdxIntegrationEvaluate the following integrals. 29. 125xx2x6dxIntegrationEvaluate the following integrals. 30. 21x2x3x212xdxIntegration Evaluate the following integrals. 31. 6x2x45x2+4dxIntegration Evaluate the following integrals. 32. 4x2x3xdxIntegration Evaluate the following integrals. 33. 3x2+4x6x23x+2dxIntegration Evaluate the following integrals. 34. 2z3+z26z+7z2+z6dzSimple linear factors Evaluate the following integrals. 23. x2+12x4x34xdx36ESimple linear factors Evaluate the following integrals. 25. dxx410x2+9Simple linear factors Evaluate the following integrals. 26. 052x24x32dxRepeated linear factors Evaluate the following integrals. 27. 81x39x2dxRepeated linear factors Evaluate the following integrals. 28. 16x2(x6)(x+2)2dxRepeated linear factors Evaluate the following integrals. 29. 11x(x+3)2dxRepeated linear factors Evaluate the following integrals. 30. dxx32x24x+8Repeated linear factors Evaluate the following integrals. 31. 2x3+x2dx