In the following exercises, verify by differentiation that ∫lnxdx=x(lnx−1)+C,∫lnxdx=x(lnx−1)+C, then use appropriate changes of variables to compute the integral.
57. ∫lnxdx∫lnxdx
(Hint:∫lnxdx=12∫xln(x2)dx)∫lnxdx=12∫xln(x2)dx))
58. ∫x2ln2xdx∫x2ln2xdx
59. ∫lnxx2dx∫lnxx2dx(Hint:Setu=1x.)(Hint:Setu=1x.)
60. ∫lnx√xdx∫lnx√xdx(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)dx∫tan(2x)dx
64. ∫sin(3x)−cos(3x)sin(3x)+cos(3x)dx∫sin(3x)−cos(3x)sin(3x)+cos(3x)dx
65. ∫xsin(x2)cos(x2)dx∫xsin(x2)cos(x2)dx
66. ∫xcsc(x2)dx∫xcsc(x2)dx
67. ∫ln(cosx)tanxdx∫ln(cosx)tanxdx
68. ∫ln(cscx)cotxdx∫ln(cscx)cotxdx
69. ∫ex−e−xex+e−xdx∫ex−e−xex+e−xdx
In the following exercises, evaluate the definite integral.
70. ∫211+2x+x23x+3x2+x3dx∫211+2x+x23x+3x2+x3dx
MISSING
In the following exercises, integrate using the indicated substitution.
71. ∫xx−100dx;u=x−100∫xx−100dx;u=x−100
72. ∫y−1y+1dy;u=y+1∫y−1y+1dy;u=y+1
73. ∫1−x23x−x3dx;u=3x−x3∫1−x23x−x3dx;u=3x−x3
74. ∫sinx+cosxsinx−cosxdx;u=sinx−cosx∫sinx+cosxsinx−cosxdx;u=sinx−cosx
75. ∫e2x√1−e2xdx;u=e2x∫e2x√1−e2xdx;u=e2x
76. ∫ln(x)√1−(lnx)2xdx;u=lnx∫ln(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=e−xy=e−x 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=−2−xy=−2−x over [0,1][0,1]
Candela Citations
- Calculus Volume 2. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-2/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction