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All Textbook Solutions for Calculus Volume 2

For the following exercises, solve each problem. 426.[T] A chain hangs from two posts four meters apart to form a catenary described by the equation y = 4 cosh(x/4) — 3. Find the total length of the catenary (arc length).For the following exercises, solve each problem. 427.[T] A high-voltage power line is a catenary described by y = 10 cosh(x/10). Find the ratio of the area under the catenary to its arc length. What do you notice?For the following exercises, solve each problem. 428. A telephone line is a catenary described by y = a cosh(x/a). Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?For the following exercises, solve each problem. 429. Prove the formula for the derivative of y = sinh-1 (x) by differentiating x = sinh(y). (Hint: Use hyperbolic trigonometric identities.)For the following exercises, solve each problem. 430. Prove the formula for the derivative of y = cosh-1 (x) by differentiating x = cosh(y). (Hint-Use hyperbolic trigonometric identities.)For the following exercises, solve each problem. 431. Prove the formula for the derivative of y = sech-1 (x) by differentiating x = sech(y). (Hint: Use hyperbolic trigonometric identities.)For the following exercises, solve each problem. 432. Prove that (cosh (a) + sinh(x))n = cosh(nx) + sinh(nx).For the following exercises, solve each problem. 433. Prove the expression for sinh-1(x). Multiply x=sinh(y)=(1/2)(eyey) by 2eyand solve for y. Does your expression match the textbook?For the following exercises, solve each problem. 434. Prove the expression for cosh-1 (x). Multiply x=cosh(y)=(1/2)(eyey) by 2eyand solve for y. Does your expression match the textbook?True or False? Justify your answer with a proof or a counterexample. 435. The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.True or False? Justify your answer with a proof or a counterexample. 436.If the force is constant, the amount of work to move an object from x = a to x = b is F(b — a).True or False? Justify your answer with a proof or a counterexample. 437.The disk method can be used in any situation in which the washer method is successful at finding the volume of a solid of revolution.True or False? Justify your answer with a proof or a counterexample. 438.If the half-life of seaborgium-266 is 360 ms, then k = (ln(2))/360.For the following exercises, use the requested method to determine the volume of the solid. 439. The volume that has a base of the ellipse x2/4+y2/9=1 and cross-sections of an equilateral triangle perpendicular to the y-axis. Use the method of slicing.For the following exercises, use the requested method to determine the volume of the solid. 440.y = x2— x, from x = 1 to x = 4, rotated around they-axis using the washer method For the following exercises, use the requested method to determine the volume of the solid. 441.x = y2and x = 3y rotated around the y-axis using the washer method For the following exercises, use the requested method to determine the volume of the solid. 442.x = 2y2- y3, x = 0, and y = 0 rotated around the x-axis using cylindrical shells For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 443.y = x3, x =0, y = 0, and x = 2 For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 444.y = x2— x and x = 0 For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 445. [T] y = ln(x) + 2 and y = x For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 446.y = x2and y=x For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 447.y = 5 + x, y = x2, x = 0, and x = 1 For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 448. Below x2+y2=1 and above y =1-x For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 449.Find the mass of =ex on a disk centered at the origin with radius 4.For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 450. Find the center of mass for =tan2x on x(4,4) .For the following exercises, find a.the area of the region, b.the volume of the solid when rotated around the x-axis, and c.the volume of the solid when rotated around the y-axis. Use whichever method seems most appropriate to you. 451.Find the mass and the center of mass of =1 on the region bounded by y = x5and y=x .For the following exercises, find the requested arc lengths. 452. The length of x for y = cosh(x) from x = 0 to x = 2.For the following exercises, find the requested arc lengths. 453. The length of y for x=3y from y = 0 to y = 4 For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis. 454.The shape created by revolving the region between y = 4 + x, y = 3 - x, x = 0, and x = 2 rotated around the y-axis.For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis. 455. The loudspeaker created by revolving y = 1/x from x = 1 to x = 4 around the x-axis.For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 m and width 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m3. 456. Find the total force on the wall of the dam.For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height 205 m and width 388 m. Assume the current depth of the water is 180 m. The density of water is 1000 kg/m3. 457.You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and 45 °F outside and the temperature of the body is 78°F. You know the cooling constant is k = 0.00824°F/min. When did the victim die, assuming that a human's temperature is 98°F?For the following exercise, consider the stock market crash in 1929 in the United States. The table lists the Dow Jones industrial average per year leading up to the crash. Years after 1920 Value ($) l 63.90 3 100 5 110 7 160 9 381.17 Source: http://stockcharts.com/ freech arts/historical/ djial9201940.html 458. [T] The best-fit exponential curve to these data is given by y=40.71+1.224x . Why do you think the gains of the market were unsustainable? Use first and second derivatives to help justify your answer. What would this model predict the Dow Jones industrial average to be in 2014?For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings. 459. Find the volume of the catenoid y = cosh(x) from x=-1 to x = 1 that is created by rotating this curve around the x-axis, as shown here.For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings. 460. Find surface area of the catenoid y = cosh(x) from x=-1 to x = 1 that is created by rotating this curve around the x-axis.In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. 1. x3e2xdx In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. 2. x3In(x)dx In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. 3. y3cosydx In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. 4. x2arctanxdx In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. 5. e3xsin(2x)dx Find the integral by using the simplest method. Not all problems require integration by parts. 6. vsinvdv Find the integral by using the simplest method. Not all problems require integration by parts. 7. Inxdx (Hint: Inxdx is equivalent to 1In(x)dx. )Find the integral by using the simplest method. Not all problems require integration by parts. 8. xcosxdx Find the integral by using the simplest method. Not all problems require integration by parts. 9. tan1xdx Find the integral by using the simplest method. Not all problems require integration by parts. 10. x2exdx Find the integral by using the simplest method. Not all problems require integration by parts. 11. xsin(2x)dx Find the integral by using the simplest method. Not all problems require integration by parts. 12. xe4xdx Find the integral by using the simplest method. Not all problems require integration by parts. 13. xexdx Find the integral by using the simplest method. Not all problems require integration by parts. 14. xcos3xdx Find the integral by using the simplest method. Not all problems require integration by parts. 15. x2cosxdx Find the integral by using the simplest method. Not all problems require integration by parts. 16. xInxdx Find the integral by using the simplest method. Not all problems require integration by parts. 17. In(2x+1)dx Find the integral by using the simplest method. Not all problems require integration by parts. 18. x2e4xdx Find the integral by using the simplest method. Not all problems require integration by parts. 19. exsinxdx Find the integral by using the simplest method. Not all problems require integration by parts. 20. excosxdx Find the integral by using the simplest method. Not all problems require integration by parts. 21. xex2dx Find the integral by using the simplest method. Not all problems require integration by parts. 22. x2exdx Find the integral by using the simplest method. Not all problems require integration by parts. 23. sin(In( 2x))dx Find the integral by using the simplest method. Not all problems require integration by parts. 24. cos(Inx)dx Find the integral by using the simplest method. Not all problems require integration by parts. 25. ( Inx)2dx Find the integral by using the simplest method. Not all problems require integration by parts. 26. In(x2)dx Find the integral by using the simplest method. Not all problems require integration by parts. 27. x2Inxdx Find the integral by using the simplest method. Not all problems require integration by parts. 28. sin1xdx Find the integral by using the simplest method. Not all problems require integration by parts. 29. cos1(2x)dx Find the integral by using the simplest method. Not all problems require integration by parts. 30. xarctanxdx Find the integral by using the simplest method. Not all problems require integration by parts. 31. x2sinxdx Find the integral by using the simplest method. Not all problems require integration by parts. 32. x3cosxdx Find the integral by using the simplest method. Not all problems require integration by parts. 33. x3sinxdx Find the integral by using the simplest method. Not all problems require integration by parts. 34. x3exdx Find the integral by using the simplest method. Not all problems require integration by parts. 35. xsec1xdx Find the integral by using the simplest method. Not all problems require integration by parts. 36. xsec2xdx Find the integral by using the simplest method. Not all problems require integration by parts. 37. xcoshxdx Compute the definite integrals. Use a graphing utility to confirm your answers. 38. 1/e1Inxdx Compute the definite integrals. Use a graphing utility to confirm your answers. 39. 01xe2xdx (Express the answer in exact form.)Compute the definite integrals. Use a graphing utility to confirm your answers. 40. 01exdx(letu=x)Compute the definite integrals. Use a graphing utility to confirm your answers. 41. 1eIn(x2)dx Compute the definite integrals. Use a graphing utility to confirm your answers. 42. 0xcosxdx Compute the definite integrals. Use a graphing utility to confirm your answers. 43. xsinxdx (Express the answer in exact form.)Compute the definite integrals. Use a graphing utility to confirm your answers. 44. 03In(x2+1)dx (Express the answer in exact form.)Compute the definite integrals. Use a graphing utility to confirm your answers. 45. 0/2x2sinxdx (Express the answer in exact form.)Compute the definite integrals. Use a graphing utility to confirm your answers. 46. 01x5xdx (Express the answer using five significant digits.)Compute the definite integrals. Use a graphing utility to confirm your answers. 47. Evaluate cosxIn(sinx)dx Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. 48. xnexdx=xnexn x n1 e xdx Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. 49. xncosxdx=xnsinxn x n1sinxdx Derive the following formulas using the technique of integration by parts. Assume that n is a positive integer. These formulas are called reduction formulas because the exponent in the x term has been reduced by one in each case. The second integral is simpler than the original integral. 50. xnsinxdx=___State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 52. xInxdx State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 53. In2xxdx State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 54. xexdx State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 55. xex23dx State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 56. x2sinxdx State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. 57. x2sin(3x3+2)dx Sketch the region bounded above by the curve, the x-axis, and x=1 , and find the area of the region. Provide the exact form or round answers to the number of places indicated. 58. y=2xex (Approximate answer to four decimal places.)Sketch the region bounded above by the curve, the x-axis, and x=1 , and find the area of the region. Provide the exact form or round answers to the number of places indicated. 59. y=exsin(x) (Approximate answer to five decimal places.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 60. y=sinx , y=0 , x=2 , x=3 about the y-axis (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 61. y=exy=0 , x=1x=0 ; about x=1 (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 62. A particle moving along a straight line has a velocity of v(t)=t2et after t sec. How far does it travel in the ?rst 2 sec? (Assume the units are in feet and express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 63. Find the area under the graph of y=sec3x from x=0 to x=1 . (Round the answer to two significant digits.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 64. Find the area between y=(x2)ex and the x-axis from x=2 to x=5 . (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 65. Find the area of the region enclosed by the curve y=xcosx and the x-axis for 112x132 . (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 66. Find the volume of the solid generated by revolving the region bounded by the curve y=Inx , the x-axis, and the vertical line x=e2 about the x-axis. (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 67. Find the volume of the solid generated by revolving the region bounded by the curve y=4cosx and the x-axis, 2x32 about the x-axis. (Express the answer in exact form.)Find the volume generated by rotating the region bounded by the given curves about the specified line. Express the answers in exact form or approximate to the number of decimal places indicated. 68. Find the volume of the solid generated by revolving the region in the first quadrant bounded by y=ex and the x-axis, from x=0 to x=In(7) , about the y-axis. (Express the answer in exact form.)Fill in the blank to make a true statement. 69. sin2x+____ =1 Fill in the blank to make a true statement. 70. sec2x1=____Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power. 71. sin2x=___Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power. 72. cos2x=_____Evaluate each of the following integrals by u-substitution. 73. sin3xcosxdx Evaluate each of the following integrals by u-substitution. 74. cosxsinxdx Evaluate each of the following integrals by u-substitution. 75. tan5(2x)sec2(2x)dx Evaluate each of the following integrals by u-substitution. 76. sin7(2x)cos(2x)dx Evaluate each of the following integrals by u-substitution. 77. tan(x2)sec2(x2)dx Evaluate each of the following integrals by u-substitution. 78. tan2xsec2xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 79. sin3xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 80. cos3xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 81. sinxcosxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 82. cos5xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 83. sin5xcos2xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 84. sin3xcos3xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 85. sinxcosxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 86. sinxcos3xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 87. secxtanxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 88. tan(5x)dx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 89. tan2xsecxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 90. tanxsec3xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 91. sec4xdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 92. cotxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 93. cscxdx Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may he done using techniques of integration learned previously.) 94. tan3x secxdx For the following exercises, find a general formula for the integrals. 95. sin2axcosaxdx For the following exercises, find a general formula for the integrals. 96. sinaxcosaxdx .Use the double-angle formulas to evaluate the following integrals. 97. 0sin2xdx Use the double-angle formulas to evaluate the following integrals. 98. 0sin4xdx Use the double-angle formulas to evaluate the following integrals. 99. cos23xdx Use the double-angle formulas to evaluate the following integrals. 100. sin2xcos2xdx Use the double-angle formulas to evaluate the following integrals. 101. sin2xdx+ cos2xdx Use the double-angle formulas to evaluate the following integrals. 102. sin2xcos2(2x)dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 103. 02cosxsin2xdx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 104. 0sin3xsin5xdx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 105. 0cos(99x)sin(101x)dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 106. cos2(3x)dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 107. 02sinxsin(2x)sin(3x)dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 108. 04cos(x/2)sin(x/2)dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 109. /6/3 cos3x sinxdx (Round this answer to three decimal places.)For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 110. /3/3 sec2x1dx For the following exercises, evaluate the definite integrals. Express answer in exact form whenever possible. 111. 0/21cos( 2x)dx Find the area of the region bounded by the graphs of the equations y=sinx , y=sin3x , x=0 , and x=2 .Find the area of the region bounded by the graphs of the equations y=cos2x , y=sin2x , x=4 , and x=4 .A particle moves in a straight line with the velocity function v(t)=sin(t)cos2(t) . Find its position function x=f(t) if f(0)=0 .Find the average value of the function f(x)=sin2xcos3x over the interval [,] .For the following exercises, solve the differential equations. 116. dydx=sin2x . The curve passes through point (0,0) .For the following exercises, solve the differential equations. 117. dyd=sin4()For the following exercises, solve the differential equations. 118. Find the length of the curve y=In(cscx) , 4x2 .For the following exercises, solve the differential equations. 119. Find the length of the curve y=In(sinx) , 3x2 .For the following exercises, solve the differential equations. 120. Find the volume generated by revolving the curve y=cos(3x) about the x-axis, 0x36 .For the following exercises, use this information: The inner product of two functions f and g over [a, b] is de?ned by f(x)g(x)=f,g=abfgdx . Two distinct functions f and g are said to be orthogonal if f,g=0 . 121. Show that {sin(2x),cos(3x)} are orthogonal over the interval [,] .For the following exercises, use this information: The inner product of two functions f and g over [a, b] is de?ned by f(x)g(x)=f,g=abfgdx . Two distinct functions f and g are said to be orthogonal if f,g=0 . 122. Evaluate sin(mx)cos(nx)dx .For the following exercises, use this information: The inner product of two functions f and g over [a, b] is de?ned by f(x)g(x)=f,g=abfgdx . Two distinct functions f and g are said to be orthogonal if f,g=0 . 123. Integrate y=tanxsec4x .For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. 124. sin456xcosxdx or sin2xcos2xdx For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. 125. tan350xsec2xdx or tan350xsecxdx Simplify the following expressions by writing each one using a single trigonometric function. 126. 44sin2 Simplify the following expressions by writing each one using a single trigonometric function. 127. 9sec29 Simplify the following expressions by writing each one using a single trigonometric function. 128. a2+a2tan2 Simplify the following expressions by writing each one using a single trigonometric function. 129. a2+a2sinh2 Simplify the following expressions by writing each one using a single trigonometric function. 130. 16cosh216 Use the technique of completing the square to express each trinomial as the square of a binomial. 131. 4x24x+1 Use the technique of completing the square to express each trinomial as the square of a binomial. 132. 2x28x+3 Use the technique of completing the square to express each trinomial as the square of a binomial. 133. x22x+4 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 134. dx 4 x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 135. dx x 2 a 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 136. 4x2dx Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 137. dx 1+9 x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 138. x2dx 1 x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 139. dxx2 1 x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 140. dx ( 1+ x 2 )2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 141. x2+9dx Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 142. x 2 25xdx Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 143. 3d 9 2 d Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 144. dx x 6 x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 145. x6x8dx Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 146. dx ( 1+ x 2 ) 3/2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 147. dx ( x 2 9 ) 3/2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 148. 1+ x 2 dxx Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 149. x2dx x 2 1 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 150. x2dxx2+4 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 151. dxx2 x 2 +1 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 152. x2dx 1+ x 2 Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable. 153. 11( 1 x 2 )3/2dx In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 154. dx x 2 1 In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 155. dxx 1 x 2 In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 156. x21dx In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 157. x 2 1x2dx In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 158. dx1x2 In the following exercises, use the substitutions x=sinh , cosh , or tanh . Express the final answers in terms of the variable x. 159. 1+ x 2 x2dx Use the technique of completing the square to evaluate the following integrals. 160. 1x26xdx Use the technique of completing the square to evaluate the following integrals. 161. 1x2+2x+1dx Use the technique of completing the square to evaluate the following integrals. 162. 1 x 2 +2x+8dx Use the technique of completing the square to evaluate the following integrals. 163. 1 x 2 +10xdx Use the technique of completing the square to evaluate the following integrals. 164. 1 x 2 +4x12dx Evaluate the integral without using calculus: 339x2dx Find the area enclosed by the ellipse x24+y29=1 .Evaluate the integral dx 1 x 2 using two different substitutions. First, let x=cos and evaluate using trigonometric substitution. Second, let x=sin and use trigonometric substitution. Are the answers the same?Evaluate the integral dxx x 2 1 using the substitution x=sec . Next, evaluate the same integral using the substitution x=csc . Show that the results are equivalent.Evaluate the integral xx2+1 using the form 1udu . Next, evaluate the same integral using x=tan . Are the results the same?State the method of integration you would use to evaluate the integral xx2+1dx . Why did you choose this method?State the method of integration you would use to evaluate the integral x2x21dx . Why did you choose this method?Evaluate 11xdxx2+1 Find the length of the arc of the curve over the specified interval: y=Inx , [1,5] . Round the answer to three decimal places.Find the surface area of the solid generated by revolving the region bounded by the graphs of y=x2 , y=0 , x=0 , and x=2 about the x-axis. (Round the answer to three decimal places).The region bounded by the graph of f(x)=11+x2 and the x-axis between x=0 and x=1 is revolved about the x-axis. Find the volume of the solid that is generated.Solve the initial-value problem for y as a function of x. 176. (x2+36)dydx=1 , y(6)=0 Solve the initial-value problem for y as a function of x. 177. (64x2)dydx=1 , y(0)=3 Solve the initial-value problem for y as a function of x. 178. Find the area bounded by y=2644x2 , x=0 , y=0 , and x=2 .An oil storage tank can he described as the volume generated by revolving the area bounded by y=1664+x2 , x=0 , y=0 , x=2 about the x-axis. Find the volume of the tank (in cubic meters).During each cycle, the velocity v (in feet per second) of a robotic welding device is given by v=2t144+t2 , where t is time in seconds. Find the expression for the displacement s (in feet) as a function of t if s=0 when t=0 .Find the length of the curve y=16x2 between x=0 and x=2 .Express the rational function as a sum or difference of two simpler rational expressions. 182. 1(x3)(x2)Express the rational function as a sum or difference of two simpler rational expressions. 183. x2+1x(x+1)(x+2)Express the rational function as a sum or difference of two simpler rational expressions. 184. 1x3x Express the rational function as a sum or difference of two simpler rational expressions. 185. 3x+1x2 Express the rational function as a sum or difference of two simpler rational expressions. 186. 3x2x2+1 (Hint: Use long division first.)Express the rational function as a sum or difference of two simpler rational expressions. 187. 2x4x22x Express the rational function as a sum or difference of two simpler rational expressions. 188. 1(x1)(x2+1)Express the rational function as a sum or difference of two simpler rational expressions. 189. 1x2(x1)Express the rational function as a sum or difference of two simpler rational expressions. 190. xx24 Express the rational function as a sum or difference of two simpler rational expressions. 191. 1x(x1)(x2)(x3)Express the rational function as a sum or difference of two simpler rational expressions. 192. 1x41=1(x+1)(x1)(x2+1)Express the rational function as a sum or difference of two simpler rational expressions. 193. 3x2x31=3x2(x1)(x2+x+1)Express the rational function as a sum or difference of two simpler rational expressions. 194. 2x(x+2)2 Express the rational function as a sum or difference of two simpler rational expressions. 195. 3x4+x3+20x2+3x+31(x+1)( x 2+4)2 Use the method of partial fractions to evaluate each of the following integrals. 196. dx( x3)( x2)Use the method of partial fractions to evaluate each of the following integrals. 197. 3xx2+2x8dx Use the method of partial fractions to evaluate each of the following integrals. 198. dxx3x Use the method of partial fractions to evaluate each of the following integrals. 199. xx24dx Use the method of partial fractions to evaluate each of the following integrals. 200. dxx( x1)( x2)( x3)Use the method of partial fractions to evaluate each of the following integrals. 201. 2x2+4x+22x2+2x+10dx Use the method of partial fractions to evaluate each of the following integrals. 202. dxx25x+6 Use the method of partial fractions to evaluate each of the following integrals. 203. 2xx2+xdx Use the method of partial fractions to evaluate each of the following integrals. 204. 2x2x6dx Use the method of partial fractions to evaluate each of the following integrals. 205. dxx32x24x+8 Use the method of partial fractions to evaluate each of the following integrals. 206. dxx410x2+9 Evaluate the following integrals, which have irreducible quadratic factors. 207. 2( x4)( x 2 +2x+6)dx Evaluate the following integrals, which have irreducible quadratic factors. 208. x2x3x2+4x4dx Evaluate the following integrals, which have irreducible quadratic factors. 209. x3+6x2+3x+6x3+2x2dx Evaluate the following integrals, which have irreducible quadratic factors. 210. x( x1) ( x 2 +2x+2 )2dx Use the method of partial fractions to evaluate the following integrals. 211. 3x+4( x 2 +4)( 3x)dx Use the method of partial fractions to evaluate the following integrals. 212. 2 ( x+2 )2( 2x)dx Use the method of partial fractions to evaluate the following integrals. 213. 3x+4x32x4dx (Hint: Use the rational root theorem.)Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 214. 01ex36e 2xdx (Give the exact answer and the decimal equivalent. Round to five decimal places.)Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 215. exdxe 2xexdx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 216. sinxdx1 cos2x Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 217. sinx cos2x+cosx6dx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 218. 1x1+xdx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 219. dt ( e t e t )2 Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 220. 1+ex1exdx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 221. dx1+ x+1 Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 222. dxx+x4 Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 223. cosxsinx( 1sinx)dx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 224. ex ( e 2x 4 )2dx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 225. 121x24 x 2dx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 226. 12+e xdx Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals. 227. 11+exdx Use the given substitution to convert the integral to an integral of a rational function, then evaluate. 228. 1tt3dtt=x3 Use the given substitution to convert the integral to an integral of a rational function, then evaluate. 229. 1x+x3dx ; x=u6 Graph the curve y=x1+x over the interval [0,5] . Then, find the area of the region bounded by the curve, the x-axis, and the line x=4 .Find the volume of the solid generated when the region bounded by y=1/x(3x) , y=0 , x=1 , and x=2 is revolved about the x-axis.The velocity of a particle moving along a line is a function of time given by v(t)=88t2t2+1 . Find the distance that the particle has traveled after t=5 .Solve the initial-value problem for x as a function of t. 233. (t27t+12)dxdt=1 , (t4,x(5)=0)Solve the initial-value problem for x as a function of t. 234. (t+5)dxdt=x2+1 , t5 , x(1)=tan1 Solve the initial-value problem for x as a function of t. 235. (2t32t2+t1)dxdt=3 , x(2)=0 Find the x-coordinate of the centroid of the area bounded by y(x29)=1 , y=0 , x=4 , and x=5 . (Round the answer to two decimal places.)Find the volume generated by revolving the area bounded by y=1x3+7x2+6xx=1 , x=7 , and y=0 about the y-axis.Find the area bounded by y=x12x28x20 , y=0 , x=2 , and x=4 . (Round the answer to the nearest hundredth.)Evaluate the integral dxx3+1 .For the following problems, use the substitutions tan(x2)=t , dx=21+t2dt , sinx=2t1+t2 , and cosx=1t21+t2 . 240. dx35sinx For the following problems, use the substitutions tan(x2)=t , dx=21+t2dt , sinx=2t1+t2 , and cosx=1t21+t2 . 241. Find the area under the curve y=11+sinx between x=0 and x= (Assume the dimensions are in inches.)For the following problems, use the substitutions tan(x2)=t , dx=21+t2dt , sinx=2t1+t2 , and cosx=1t21+t2 . 242. Given tan(x2)=t , derive the formulas dx=21+t2dt , sinx=2t1+t2 , and cosx=1t21+t2 .For the following problems, use the substitutions tan(x2)=t , dx=21+t2dt , sinx=2t1+t2 , and cosx=1t21+t2 . 243. Evaluate x83xdx .Use a table of integrals to evaluate the following integrals. 244. 04x1+2xdx Use a table of integrals to evaluate the following integrals. 245. x+3x2+2x+2dx Use a table of integrals to evaluate the following integrals. 246. x31+2x2dx Use a table of integrals to evaluate the following integrals. 247. 1 x 2 +6xdx Use a table of integrals to evaluate the following integrals. 248. xx+1dx Use a table of integrals to evaluate the following integrals. 249. x2x2dx Use a table of integrals to evaluate the following integrals. 250. 14x2+25dx Use a table of integrals to evaluate the following integrals. 251. dy 4 y 2 Use a table of integrals to evaluate the following integrals. 252. sin3(2x)cos(2x)dx Use a table of integrals to evaluate the following integrals. 253. csc(2w)cot(2w)dw Use a table of integrals to evaluate the following integrals. 254. 2ydy Use a table of integrals to evaluate the following integrals. 255. 013xdx x 2 +8 Use a table of integrals to evaluate the following integrals. 256. 1/41/4sec2(x)tan(x)dx Use a table of integrals to evaluate the following integrals. 257. 0/2tan2(x2)dx Use a table of integrals to evaluate the following integrals. 258. cos3xdx Use a table of integrals to evaluate the following integrals. 259. tan5(3x)dx Use a table of integrals to evaluate the following integrals. 260. sin2ycos3ydy Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 261. [T] dw1+sec( w 2 )Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 262. [T] dw1cos( 7w)Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 263. [T] 0tdt4cost+3sint Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 264. [T] x 2 93xdx Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 265. [T] dxx 1/2 +x 1/3 Use a CAS to evaluate the following integrals. Tables can also be used to verify the answer. 266. [T] dxx x1

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