Problem Set: Integrals Involving Exponential and Logarithmic Functions

In the following exercises, verify by differentiation that lnxdx=x(lnx1)+C,lnxdx=x(lnx1)+C, then use appropriate changes of variables to compute the integral.

57. lnxdxlnxdx

(Hint:lnxdx=12xln(x2)dx)lnxdx=12xln(x2)dx))

58. x2ln2xdxx2ln2xdx

59. lnxx2dxlnxx2dx(Hint:Setu=1x.)(Hint:Setu=1x.)

60. lnxxdxlnxxdx(Hint:Setu=x.)(Hint:Setu=x.)

61. Write an integral to express the area under the graph of y=1ty=1t from t=1t=1 to ex and evaluate the integral.

62. Write an integral to express the area under the graph of y=ety=et between t=0t=0 and t=lnx,t=lnx, and evaluate the integral.

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.

63. tan(2x)dxtan(2x)dx

64. sin(3x)cos(3x)sin(3x)+cos(3x)dxsin(3x)cos(3x)sin(3x)+cos(3x)dx

65. xsin(x2)cos(x2)dxxsin(x2)cos(x2)dx

66. xcsc(x2)dxxcsc(x2)dx

67. ln(cosx)tanxdxln(cosx)tanxdx

68. ln(cscx)cotxdxln(cscx)cotxdx

69. exexex+exdxexexex+exdx

In the following exercises, evaluate the definite integral.

70. 211+2x+x23x+3x2+x3dx211+2x+x23x+3x2+x3dx

MISSING

In the following exercises, integrate using the indicated substitution.

71. xx100dx;u=x100xx100dx;u=x100

72. y1y+1dy;u=y+1y1y+1dy;u=y+1

73. 1x23xx3dx;u=3xx31x23xx3dx;u=3xx3

74. sinx+cosxsinxcosxdx;u=sinxcosxsinx+cosxsinxcosxdx;u=sinxcosx

75. e2x1e2xdx;u=e2xe2x1e2xdx;u=e2x

76. ln(x)1(lnx)2xdx;u=lnxln(x)1(lnx)2xdx;u=lnx

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R50 and solve for the exact area.

37. [T] y=exy=ex over [0,1][0,1]

38. [T] y=exy=ex over [0,1][0,1]

39. [T] y=ln(x)y=ln(x) over [1,2][1,2]

40. [T] y=x+1x2+2x+6y=x+1x2+2x+6 over [0,1][0,1]

41. [T] y=2xy=2x over [1,0][1,0]

42. [T] y=2xy=2x over [0,1][0,1]