{"id":1208,"date":"2021-06-30T17:02:10","date_gmt":"2021-06-30T17:02:10","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-integrals-exponential-functions-and-logarithms\/"},"modified":"2021-11-17T02:19:51","modified_gmt":"2021-11-17T02:19:51","slug":"problem-set-integrals-exponential-functions-and-logarithms","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/problem-set-integrals-exponential-functions-and-logarithms\/","title":{"raw":"Problem Set: Integrals, Exponential Functions, and Logarithms","rendered":"Problem Set: Integrals, Exponential Functions, and Logarithms"},"content":{"raw":"

For the following exercises (1-3), find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]y=\\text{ln}(2x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794097576\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794097576\"]\r\n

[latex]\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2.<\/strong> [latex]y=\\text{ln}(2x+1)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]y=\\dfrac{1}{\\text{ln}x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793546866\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793546866\"]\r\n

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

For the following exercises (4-5), find the indefinite integral.<\/p>\r\n\r\n

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4.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dt}{3t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1+x}[\/latex]<\/p>\r\n[reveal-answer q=\"404686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"404686\"]\r\n\r\n[latex]\\text{ln}(x+1)+C[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (6-15), find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/p>\r\n\r\n

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6. [T]<\/strong> [latex]y=\\dfrac{\\text{ln}(x)}{x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. [T]<\/strong> [latex]y=x\\text{ln}(x)[\/latex]<\/p>\r\n[reveal-answer q=\"23354778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"23354778\"]\r\n\r\n[latex]\\text{ln}(x)+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [T]<\/strong> [latex]y={\\text{log}}_{10}x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [T]<\/strong> [latex]y=\\text{ln}( \\sin x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793776894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793776894\"]\r\n

[latex] \\cot (x)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. [T]<\/strong> [latex]y=\\text{ln}(\\text{ln}x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [T]<\/strong> [latex]y=7\\text{ln}(4x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793293693\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793293693\"]\r\n

[latex]\\frac{7}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [T]<\/strong> [latex]y=\\text{ln}({(4x)}^{7})[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [T]<\/strong> [latex]y=\\text{ln}( \\tan x)[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793557826\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793557826\"]\r\n

[latex] \\csc (x) \\sec x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [T]<\/strong> [latex]y=\\text{ln}( \\tan (3x))[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [T]<\/strong> [latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/p>\r\n[reveal-answer q=\"277155388\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"277155388\"]\r\n

[latex]-2 \\tan x[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (16-25), find the definite or indefinite integral.<\/p>\r\n\r\n

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\r\n\r\n16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{3+x}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dt}{3+2t}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793553651\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793553651\"]\r\n

[latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{xdx}{{x}^{2}+1}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19.\u00a0\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{{x}^{3}dx}{{x}^{2}+1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793952145\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793952145\"]\r\n

[latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{x\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/p>\r\n[reveal-answer q=\"37762200\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"37762200\"]\r\n\r\n[latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\cos xdx}{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794005215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794005215\"][latex]\\frac{1}{2}\\text{ln}(2)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\cot (3x)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{(\\text{ln}x)}^{2}dx}{x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793421199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793421199\"]\r\n

[latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises (26-35), compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]\r\n

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\r\n\r\n26.\u00a0<\/strong>[latex]y=\\sqrt{{x}^{2}+1}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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27.\u00a0<\/strong>[latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793595181\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793595181\"]\r\n

[latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28.\u00a0<\/strong>[latex]y={e}^{ \\sin x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]y={x}^{-1\\text{\/}x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793465231\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793465231\"]\r\n

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

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30.\u00a0<\/strong>[latex]y={e}^{(ex)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n31.<\/strong> [latex]y={x}^{e}[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1167793445732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793445732\"]\r\n

[latex]e{x}^{e-1}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0<\/strong>[latex]y={x}^{(ex)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>[latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{x}[\/latex]<\/p>\r\n\r\n

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1<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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34.<\/strong> [latex]y={x}^{-1\\text{\/}\\text{ln}x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35.\u00a0<\/strong>[latex]y={e}^{\\text{\u2212}\\text{ln}x}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793315539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793315539\"]\r\n

[latex]-\\frac{1}{{x}^{2}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (36-40), evaluate by any method.<\/p>\r\n\r\n

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36.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{5}^{10}\\frac{dt}{t}-{\\displaystyle\\int }_{5x}^{10x}\\frac{dt}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\displaystyle\\int }_{-2}^{-1}\\frac{dx}{x}[\/latex]\r\n\r\n[reveal-answer q=\"fs-id1167793455346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793455346\"]\r\n

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

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38.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{1}\\frac{dt}{t}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167794146824\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794146824\"]\r\n

[latex]\\frac{1}{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}( \\sec x+ \\tan x)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (41-, use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/p>\r\n\r\n

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41.\u00a0<\/strong>Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{ln}x.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793541846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793541846\"]\r\n

[latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. [T]<\/strong> Find the arc length of [latex]\\text{ln}x[\/latex] from [latex]x=1[\/latex] to [latex]x=2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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43.\u00a0<\/strong>Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=2.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793713050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793713050\"]\r\n

[latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0<\/strong>Find the volume of the shape created when rotating this curve from [latex]x=1\\text{ to }x=2[\/latex] around the [latex]x[\/latex]-axis, as pictured here.<\/p>\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. [T]<\/strong> Find the surface area of the shape created when rotating the curve in the previous exercise from [latex]x=1[\/latex] to [latex]x=2[\/latex] around the [latex]x[\/latex]-axis.<\/p>\r\n[reveal-answer q=\"fs-id1167793510687\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793510687\"]\r\n

2.8656<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

If you are unable to find intersection points analytically in the following exercises (46-48), use a calculator.<\/p>\r\n\r\n

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46.\u00a0<\/strong>Find the area of the hyperbolic quarter-circle enclosed by [latex]x=2\\text{ and }y=2[\/latex] above [latex]y=\\frac{1}{x}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. [T]<\/strong> Find the arc length of [latex]y=\\frac{1}{x}[\/latex] from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\r\n[reveal-answer q=\"fs-id1167793570742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167793570742\"]\r\n

3.1502<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>Find the area under [latex]y=\\frac{1}{x}[\/latex] and above the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises (49-53), verify the derivatives and antiderivatives.<\/p>\r\n\r\n

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49.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\dfrac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{x-a}{x+a}\\right)=\\dfrac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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51.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{1+\\sqrt{1-{x}^{2}}}{x}\\right)=-\\dfrac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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52.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\dfrac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>","rendered":"

For the following exercises (1-3), find the derivative [latex]\\frac{dy}{dx}.[\/latex]<\/p>\n

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1.\u00a0<\/strong>[latex]y=\\text{ln}(2x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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2.<\/strong> [latex]y=\\text{ln}(2x+1)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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3.\u00a0<\/strong>[latex]y=\\dfrac{1}{\\text{ln}x}[\/latex]<\/p>\n

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

For the following exercises (4-5), find the indefinite integral.<\/p>\n

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4.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dt}{3t}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1+x}[\/latex]<\/p>\n

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

For the following exercises (6-15), find the derivative [latex]dy\\text{\/}dx.[\/latex] (You can use a calculator to plot the function and the derivative to confirm that it is correct.)<\/p>\n

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6. [T]<\/strong> [latex]y=\\dfrac{\\text{ln}(x)}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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7. [T]<\/strong> [latex]y=x\\text{ln}(x)[\/latex]<\/p>\n

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

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8. [T]<\/strong> [latex]y={\\text{log}}_{10}x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [T]<\/strong> [latex]y=\\text{ln}( \\sin x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\cot (x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [T]<\/strong> [latex]y=\\text{ln}(\\text{ln}x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [T]<\/strong> [latex]y=7\\text{ln}(4x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\frac{7}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [T]<\/strong> [latex]y=\\text{ln}({(4x)}^{7})[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [T]<\/strong> [latex]y=\\text{ln}( \\tan x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]\\csc (x) \\sec x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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14. [T]<\/strong> [latex]y=\\text{ln}( \\tan (3x))[\/latex]<\/p>\n<\/div>\n<\/div>\n

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15. [T]<\/strong> [latex]y=\\text{ln}({ \\cos }^{2}x)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
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[latex]-2 \\tan x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (16-25), find the definite or indefinite integral.<\/p>\n

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16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dx}{3+x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{dt}{3+2t}[\/latex]<\/p>\n

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[latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{xdx}{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19.\u00a0\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{{x}^{3}dx}{{x}^{2}+1}[\/latex]<\/p>\n

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[latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{x\\text{ln}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{2}^{e}\\dfrac{dx}{{(x\\text{ln}(x))}^{2}}[\/latex]<\/p>\n

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[latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{ \\cos xdx}{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4} \\tan xdx[\/latex]<\/p>\n

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[latex]\\frac{1}{2}\\text{ln}(2)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\cot (3x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{(\\text{ln}x)}^{2}dx}{x}[\/latex]<\/p>\n

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[latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (26-35), compute [latex]dy\\text{\/}dx[\/latex] by differentiating [latex]\\text{ln}y.[\/latex]<\/p>\n

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26.\u00a0<\/strong>[latex]y=\\sqrt{{x}^{2}+1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27.\u00a0<\/strong>[latex]y=\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}[\/latex]<\/p>\n

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[latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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28.\u00a0<\/strong>[latex]y={e}^{ \\sin x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>[latex]y={x}^{-1\\text{\/}x}[\/latex]<\/p>\n

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[latex]{x}^{-2-(1\\text{\/}x)}(\\text{ln}x-1)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30.\u00a0<\/strong>[latex]y={e}^{(ex)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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31.<\/strong> [latex]y={x}^{e}[\/latex]<\/p>\n

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[latex]e{x}^{e-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32.\u00a0<\/strong>[latex]y={x}^{(ex)}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33.\u00a0<\/strong>[latex]y=\\sqrt{x}\\sqrt[3]{x}\\sqrt[6]{x}[\/latex]<\/p>\n

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1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34.<\/strong> [latex]y={x}^{-1\\text{\/}\\text{ln}x}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35.\u00a0<\/strong>[latex]y={e}^{\\text{\u2212}\\text{ln}x}[\/latex]<\/p>\n

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[latex]-\\frac{1}{{x}^{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises (36-40), evaluate by any method.<\/p>\n

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36.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{5}^{10}\\frac{dt}{t}-{\\displaystyle\\int }_{5x}^{10x}\\frac{dt}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{{e}^{\\pi }}\\frac{dx}{x}+{\\displaystyle\\int }_{-2}^{-1}\\frac{dx}{x}[\/latex]<\/p>\n

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[latex]\\pi -\\text{ln}(2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{1}\\frac{dt}{t}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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39.\u00a0<\/strong>[latex]\\frac{d}{dx}{\\displaystyle\\int }_{x}^{{x}^{2}}\\frac{dt}{t}[\/latex]<\/p>\n

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[latex]\\frac{1}{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}( \\sec x+ \\tan x)[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (41-, use the function [latex]\\text{ln}x.[\/latex] If you are unable to find intersection points analytically, use a calculator.<\/p>\n

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41.\u00a0<\/strong>Find the area of the region enclosed by [latex]x=1[\/latex] and [latex]y=5[\/latex] above [latex]y=\\text{ln}x.[\/latex]<\/p>\n

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[latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42. [T]<\/strong> Find the arc length of [latex]\\text{ln}x[\/latex] from [latex]x=1[\/latex] to [latex]x=2.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43.\u00a0<\/strong>Find the area between [latex]\\text{ln}x[\/latex] and the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=2.[\/latex]<\/p>\n

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[latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44.\u00a0<\/strong>Find the volume of the shape created when rotating this curve from [latex]x=1\\text{ to }x=2[\/latex] around the [latex]x[\/latex]-axis, as pictured here.<\/p>\n

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45. [T]<\/strong> Find the surface area of the shape created when rotating the curve in the previous exercise from [latex]x=1[\/latex] to [latex]x=2[\/latex] around the [latex]x[\/latex]-axis.<\/p>\n

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2.8656<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

If you are unable to find intersection points analytically in the following exercises (46-48), use a calculator.<\/p>\n

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46.\u00a0<\/strong>Find the area of the hyperbolic quarter-circle enclosed by [latex]x=2\\text{ and }y=2[\/latex] above [latex]y=\\frac{1}{x}.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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47. [T]<\/strong> Find the arc length of [latex]y=\\frac{1}{x}[\/latex] from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\n

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3.1502<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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48.\u00a0<\/strong>Find the area under [latex]y=\\frac{1}{x}[\/latex] and above the [latex]x[\/latex]-axis from [latex]x=1\\text{ to }x=4.[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises (49-53), verify the derivatives and antiderivatives.<\/p>\n

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49.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}+1})=\\dfrac{1}{\\sqrt{1+{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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50.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{x-a}{x+a}\\right)=\\dfrac{2a}{({x}^{2}-{a}^{2})}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}\\left(\\dfrac{1+\\sqrt{1-{x}^{2}}}{x}\\right)=-\\dfrac{1}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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52.\u00a0<\/strong>[latex]\\frac{d}{dx}\\text{ln}(x+\\sqrt{{x}^{2}-{a}^{2}})=\\dfrac{1}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\text{ln}(x)\\text{ln}(\\text{ln}x)}=\\text{ln}(\\text{ln}(\\text{ln}x))+C[\/latex]<\/p>\n<\/div>\n<\/div>\n\n\t\t\t

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