{"id":1157,"date":"2021-06-30T17:02:02","date_gmt":"2021-06-30T17:02:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-integrals-involving-exponential-and-logarithmic-functions\/"},"modified":"2021-11-17T01:51:12","modified_gmt":"2021-11-17T01:51:12","slug":"problem-set-integrals-involving-exponential-and-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-integrals-involving-exponential-and-logarithmic-functions\/","title":{"raw":"Problem Set: Integrals Involving Exponential and Logarithmic Functions","rendered":"Problem Set: Integrals Involving Exponential and Logarithmic Functions"},"content":{"raw":"In the following exercises, verify by differentiation that [latex]\\displaystyle\\int \\text{ln}xdx=x(\\text{ln}x-1)+C,[\/latex] then use appropriate changes of variables to compute the integral.\r\n
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\r\n\r\n57.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}xdx[\/latex]\r\n\r\n(Hint:<\/em>[latex]\\displaystyle\\int \\text{ln}xdx=\\frac{1}{2}\\displaystyle\\int x\\text{ln}({x}^{2})dx)[\/latex])<\/em>\r\n\r\n<\/div>\r\n<\/div>\r\n
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58.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}{\\text{ln}}^{2}xdx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572399000\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572399000\"]\r\n

[latex]\\frac{1}{9}{x}^{3}(\\text{ln}({x}^{3})-1)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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59.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{ln}x}{{x}^{2}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\frac{1}{x}\\text{.})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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60.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\sqrt{x}\\text{.})[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572168712\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572168712\"]\r\n

[latex]2\\sqrt{x}(\\text{ln}x-2)+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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61.\u00a0<\/strong>Write an integral to express the area under the graph of [latex]y=\\frac{1}{t}[\/latex] from [latex]t=1[\/latex] to ex<\/sup><\/em> and evaluate the integral.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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62.\u00a0<\/strong>Write an integral to express the area under the graph of [latex]y={e}^{t}[\/latex] between [latex]t=0[\/latex] and [latex]t=\\text{ln}x,[\/latex] and evaluate the integral.<\/p>\r\n[reveal-answer q=\"fs-id1170571689752\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571689752\"]\r\n

[latex]{\\displaystyle\\int }_{0}^{\\text{ln}x}{e}^{t}dt={e}^{t}{|}_{0}^{\\text{ln}x}={e}^{\\text{ln}x}-{e}^{0}=x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.<\/p>\r\n\r\n

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63.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan (2x)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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64.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\sin (3x)- \\cos (3x)}{ \\sin (3x)+ \\cos (3x)}dx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571572016\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571572016\"]\r\n

[latex]-\\frac{1}{3}\\text{ln}( \\sin (3x)+ \\cos (3x))[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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65.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x \\sin ({x}^{2})}{ \\cos ({x}^{2})}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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66.\u00a0<\/strong>[latex]\\displaystyle\\int x \\csc ({x}^{2})dx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571777931\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571777931\"]\r\n

[latex]-\\frac{1}{2}\\text{ln}| \\csc ({x}^{2})+ \\cot ({x}^{2})|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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67.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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68.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}( \\csc x) \\cot xdx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571613792\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571613792\"]\r\n

[latex]-\\frac{1}{2}{(\\text{ln}( \\csc x))}^{2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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69.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{{e}^{x}+{e}^{\\text{\u2212}x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

In the following exercises, evaluate the definite integral.<\/p>\r\n\r\n

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70.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{2}\\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"fs-id1170572347069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572347069\"][latex]\\frac{1}{3}\\text{ln}(\\frac{26}{7})[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\nMISSING<\/span><\/strong>\r\n

In the following exercises, integrate using the indicated substitution.<\/p>\r\n\r\n

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71.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{x-100}dx;u=x-100[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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72.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{y-1}{y+1}dy;u=y+1[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572627138\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572627138\"]\r\n

[latex]y-2\\text{ln}|y+1|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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73.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n74.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}dx;u= \\sin x- \\cos x[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1170571712850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571712850\"]\r\n

[latex]\\text{ln}| \\sin x- \\cos x|+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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75.\u00a0<\/strong>[latex]\\displaystyle\\int {e}^{2x}\\sqrt{1-{e}^{2x}}dx;u={e}^{2x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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76.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}(x)\\frac{\\sqrt{1-{(\\text{ln}x)}^{2}}}{x}dx;u=\\text{ln}x[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572379266\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572379266\"]\r\n

[latex]-\\frac{1}{3}{(1-(\\text{ln}{x}^{2}))}^{3\\text{\/}2}+C[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R<\/em>50<\/sub> and solve for the exact area.<\/p>\r\n\r\n

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37. [T]<\/strong> [latex]y={e}^{x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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38. [T]<\/strong> [latex]y={e}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170572129108\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170572129108\"]\r\n

Exact solution: [latex]\\frac{e-1}{e},{R}_{50}=0.6258.[\/latex] Since [latex]f[\/latex] is decreasing, the right endpoint estimate underestimates the area.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. [T]<\/strong> [latex]y=\\text{ln}(x)[\/latex] over [latex]\\left[1,2\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. [T]<\/strong> [latex]y=\\frac{x+1}{{x}^{2}+2x+6}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571568951\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571568951\"]\r\n

Exact solution: [latex]\\frac{2\\text{ln}(3)-\\text{ln}(6)}{2},{R}_{50}=0.2033.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. [T]<\/strong> [latex]y={2}^{x}[\/latex] over [latex]\\left[-1,0\\right][\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [T]<\/strong> [latex]y=\\text{\u2212}{2}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1170571628910\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1170571628910\"]\r\n

Exact solution: [latex]-\\frac{1}{\\text{ln}(4)},{R}_{50}=-0.7164.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

In the following exercises, verify by differentiation that [latex]\\displaystyle\\int \\text{ln}xdx=x(\\text{ln}x-1)+C,[\/latex] then use appropriate changes of variables to compute the integral.<\/p>\n

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57.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}xdx[\/latex]<\/p>\n

(Hint:<\/em>[latex]\\displaystyle\\int \\text{ln}xdx=\\frac{1}{2}\\displaystyle\\int x\\text{ln}({x}^{2})dx)[\/latex])<\/em><\/p>\n<\/div>\n<\/div>\n

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58.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}{\\text{ln}}^{2}xdx[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{9}{x}^{3}(\\text{ln}({x}^{3})-1)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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59.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{ln}x}{{x}^{2}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\frac{1}{x}\\text{.})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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60.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex][latex](Hint\\text{:}\\text{Set}u=\\sqrt{x}\\text{.})[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]2\\sqrt{x}(\\text{ln}x-2)+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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61.\u00a0<\/strong>Write an integral to express the area under the graph of [latex]y=\\frac{1}{t}[\/latex] from [latex]t=1[\/latex] to ex<\/sup><\/em> and evaluate the integral.<\/p>\n<\/div>\n<\/div>\n

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62.\u00a0<\/strong>Write an integral to express the area under the graph of [latex]y={e}^{t}[\/latex] between [latex]t=0[\/latex] and [latex]t=\\text{ln}x,[\/latex] and evaluate the integral.<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]{\\displaystyle\\int }_{0}^{\\text{ln}x}{e}^{t}dt={e}^{t}{|}_{0}^{\\text{ln}x}={e}^{\\text{ln}x}-{e}^{0}=x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.<\/p>\n

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63.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan (2x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

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64.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\sin (3x)- \\cos (3x)}{ \\sin (3x)+ \\cos (3x)}dx[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-\\frac{1}{3}\\text{ln}( \\sin (3x)+ \\cos (3x))[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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65.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x \\sin ({x}^{2})}{ \\cos ({x}^{2})}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

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66.\u00a0<\/strong>[latex]\\displaystyle\\int x \\csc ({x}^{2})dx[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-\\frac{1}{2}\\text{ln}| \\csc ({x}^{2})+ \\cot ({x}^{2})|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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67.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}( \\cos x) \\tan xdx[\/latex]<\/p>\n<\/div>\n<\/div>\n

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68.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}( \\csc x) \\cot xdx[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-\\frac{1}{2}{(\\text{ln}( \\csc x))}^{2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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69.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}-{e}^{\\text{\u2212}x}}{{e}^{x}+{e}^{\\text{\u2212}x}}dx[\/latex]<\/p>\n<\/div>\n

In the following exercises, evaluate the definite integral.<\/p>\n

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70.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{2}\\frac{1+2x+{x}^{2}}{3x+3{x}^{2}+{x}^{3}}dx[\/latex]<\/p>\n<\/div>\n

Show Solution<\/span><\/p>\n
[latex]\\frac{1}{3}\\text{ln}(\\frac{26}{7})[\/latex]<\/div>\n<\/div>\n<\/div>\n

MISSING<\/span><\/strong><\/p>\n

In the following exercises, integrate using the indicated substitution.<\/p>\n

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71.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{x-100}dx;u=x-100[\/latex]<\/p>\n<\/div>\n<\/div>\n

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72.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{y-1}{y+1}dy;u=y+1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]y-2\\text{ln}|y+1|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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73.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1-{x}^{2}}{3x-{x}^{3}}dx;u=3x-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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74.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}dx;u= \\sin x- \\cos x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\text{ln}| \\sin x- \\cos x|+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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75.\u00a0<\/strong>[latex]\\displaystyle\\int {e}^{2x}\\sqrt{1-{e}^{2x}}dx;u={e}^{2x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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76.\u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}(x)\\frac{\\sqrt{1-{(\\text{ln}x)}^{2}}}{x}dx;u=\\text{ln}x[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-\\frac{1}{3}{(1-(\\text{ln}{x}^{2}))}^{3\\text{\/}2}+C[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R<\/em>50<\/sub> and solve for the exact area.<\/p>\n

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37. [T]<\/strong> [latex]y={e}^{x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

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38. [T]<\/strong> [latex]y={e}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Exact solution: [latex]\\frac{e-1}{e},{R}_{50}=0.6258.[\/latex] Since [latex]f[\/latex] is decreasing, the right endpoint estimate underestimates the area.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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39. [T]<\/strong> [latex]y=\\text{ln}(x)[\/latex] over [latex]\\left[1,2\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

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40. [T]<\/strong> [latex]y=\\frac{x+1}{{x}^{2}+2x+6}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Exact solution: [latex]\\frac{2\\text{ln}(3)-\\text{ln}(6)}{2},{R}_{50}=0.2033.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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41. [T]<\/strong> [latex]y={2}^{x}[\/latex] over [latex]\\left[-1,0\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n

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42. [T]<\/strong> [latex]y=\\text{\u2212}{2}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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Exact solution: [latex]-\\frac{1}{\\text{ln}(4)},{R}_{50}=-0.7164.[\/latex] Since [latex]f[\/latex] is increasing, the right endpoint estimate overestimates the area (the actual area is a larger negative number).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>