{"id":64,"date":"2021-03-25T02:20:49","date_gmt":"2021-03-25T02:20:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/trigonometric-substitution-2\/"},"modified":"2021-11-17T02:37:46","modified_gmt":"2021-11-17T02:37:46","slug":"trigonometric-substitution-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/trigonometric-substitution-2\/","title":{"raw":"Problem Set: Trigonometric Substitution","rendered":"Problem Set: Trigonometric Substitution"},"content":{"raw":"

Simplify the following expressions by writing each one using a single trigonometric function.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]4 - 4{\\sin}^{2}\\theta [\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>[latex]9{\\sec}^{2}\\theta -9[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"971246\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"971246\"][latex]9{\\tan}^{2}\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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3.\u00a0<\/strong>[latex]{a}^{2}+{a}^{2}{\\tan}^{2}\\theta [\/latex]<\/div>\r\n
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\r\n\r\n4.\u00a0<\/strong>[latex]{a}^{2}+{a}^{2}{\\text{sinh}}^{2}\\theta [\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"640385\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"640385\"][latex]{a}^{2}{\\text{cosh}}^{2}\\theta [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]16{\\text{cosh}}^{2}\\theta -16[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the technique of completing the square to express each trinomial as the square of a binomial.<\/p>\r\n\r\n

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6.\u00a0<\/strong>[latex]4{x}^{2}-4x+1[\/latex]<\/p>\r\n[reveal-answer q=\"5785\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"5785\"][latex]4{\\left(x-\\frac{1}{2}\\right)}^{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n

<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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7.\u00a0<\/strong>[latex]2{x}^{2}-8x+3[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>[latex]\\text{-}{x}^{2}-2x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"999199\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"999199\"][latex]\\text{-}{\\left(x+1\\right)}^{2}+5[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{4-{x}^{2}}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"123893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"123893\"][latex]\\text{ln}|x+\\sqrt{\\text{-}{a}^{2}+{x}^{2}}|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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11.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{4-{x}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+9{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"649169\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"649169\"][latex]\\frac{1}{3}\\text{ln}|\\sqrt{9{x}^{2}+1}+3x|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n[reveal-answer q=\"747063\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"747063\"][latex]-\\frac{\\sqrt{1-{x}^{2}}}{x}+C[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left(1+{x}^{2}\\right)}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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16.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{2}+9}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"547820\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"547820\"][latex]9\\left[\\frac{x\\sqrt{{x}^{2}+9}}{18}+\\frac{1}{2}ln|\\frac{\\sqrt{{x}^{2}+9}}{3}+\\frac{x}{3}|\\right]+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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17.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-25}}{x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{\\theta }^{3}d\\theta }{\\sqrt{9-{\\theta }^{2}}}d\\theta [\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"888117\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"888117\"][latex]-\\frac{1}{3}\\sqrt{9-{\\theta }^{2}}\\left(18+{\\theta }^{2}\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

19.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{6}-{x}^{2}}}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{6}-{x}^{8}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"843302\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843302\"][latex]\\frac{\\left(-1+{x}^{2}\\right)\\left(2+3{x}^{2}\\right)\\sqrt{{x}^{6}-{x}^{8}}}{15{x}^{3}}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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21.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left(1+{x}^{2}\\right)}^{\\frac{3}{2}}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left({x}^{2}-9\\right)}^{\\frac{3}{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"633015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"633015\"][latex]-\\frac{x}{9\\sqrt{-9+{x}^{2}}}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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23.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}dx}{x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"321156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"321156\"][latex]\\frac{1}{2}\\left(\\text{ln}|x+\\sqrt{{x}^{2}-1}|+x\\sqrt{{x}^{2}-1}\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{{x}^{2}+4}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"32737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"32737\"][latex]-\\frac{\\sqrt{1+{x}^{2}}}{x}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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27.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{1+{x}^{2}}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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28.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}{\\left(1-{x}^{2}\\right)}^{\\frac{3}{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"402810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"402810\"][latex]\\frac{1}{8}\\left(x\\left(5 - 2{x}^{2}\\right)\\sqrt{1-{x}^{2}}+3\\text{arcsin}x\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

In the following exercises, use the substitutions [latex]x=\\text{sinh}\\theta ,\\text{cosh}\\theta [\/latex], or [latex]\\text{tanh}\\theta [\/latex]. Express the final answers in terms of the variable <\/span>x.<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-1}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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30.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"667543\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"667543\"][latex]\\text{ln}x-\\text{ln}|1+\\sqrt{1-{x}^{2}}|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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31.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{2}-1}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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32.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-1}}{{x}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"240850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"240850\"][latex]-\\frac{\\sqrt{-1+{x}^{2}}}{x}+\\text{ln}|x+\\sqrt{-1+{x}^{2}}|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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33.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1-{x}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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34.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}}{{x}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"7453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"7453\"][latex]-\\frac{\\sqrt{1+{x}^{2}}}{x}+\\text{arcsinh}x+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the technique of completing the square to evaluate the following integrals.<\/p>\r\n\r\n

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35.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}-6x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}+2x+1}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"188708\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188708\"][latex]-\\frac{1}{1+x}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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37.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+2x+8}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+10x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"410509\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"410509\"][latex]\\frac{2\\sqrt{-10+x}\\sqrt{x}\\text{ln}|\\sqrt{-10+x}+\\sqrt{x}|}{\\sqrt{\\left(10-x\\right)x}}+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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39.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{{x}^{2}+4x - 12}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40.\u00a0<\/strong>Evaluate the integral without using calculus: [latex]{\\displaystyle\\int }_{-3}^{3}\\sqrt{9-{x}^{2}}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"825487\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"825487\"][latex]\\frac{9\\pi }{2}[\/latex]; area of a semicircle with radius 3[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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41.\u00a0<\/strong>Find the area enclosed by the ellipse [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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42.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex] using two different substitutions. First, let [latex]x=\\cos\\theta [\/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\\sin\\theta [\/latex] and use trigonometric substitution. Are the answers the same?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"345067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345067\"][latex]\\text{arcsin}\\left(x\\right)+C[\/latex] is the common answer.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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43.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{{x}^{2}-1}}[\/latex] using the substitution [latex]x=\\sec\\theta [\/latex]. Next, evaluate the same integral using the substitution [latex]x=\\csc\\theta [\/latex]. Show that the results are equivalent.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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44.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+1}dx[\/latex] using the form [latex]\\displaystyle\\int \\frac{1}{u}du[\/latex]. Next, evaluate the same integral using [latex]x=\\tan\\theta [\/latex]. Are the results the same?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"728845\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"728845\"][latex]\\frac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex] is the result using either method.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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45.\u00a0<\/strong>State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int x\\sqrt{{x}^{2}+1}dx[\/latex]. Why did you choose this method?<\/div>\r\n<\/div>\r\n<\/div>\r\n
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46.\u00a0<\/strong>State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int {x}^{2}\\sqrt{{x}^{2}-1}dx[\/latex]. Why did you choose this method?<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"661937\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"661937\"]Use trigonometric substitution. Let [latex]x=\\sec\\left(\\theta \\right)[\/latex].[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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47.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{-1}^{1}\\frac{xdx}{{x}^{2}+1}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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48.\u00a0<\/strong>Find the length of the arc of the curve over the specified interval: [latex]y=\\text{ln}x,\\left[1,5\\right][\/latex]. Round the answer to three decimal places.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"256417\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"256417\"]4.367[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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49.\u00a0<\/strong>Find the surface area of the solid generated by revolving the region bounded by the graphs of [latex]y={x}^{2},y=0,x=0,\\text{and }x=\\sqrt{2}[\/latex] about the x<\/em>-axis. (Round the answer to three decimal places).<\/div>\r\n<\/div>\r\n<\/div>\r\n
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50.\u00a0<\/strong>The region bounded by the graph of [latex]f\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex] and the x<\/em>-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex] is revolved about the x-<\/em>axis. Find the volume of the solid that is generated.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"585694\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"585694\"][latex]\\frac{{\\pi }^{2}}{8}+\\frac{\\pi }{4}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Solve the initial-value problem for y<\/em> as a function of x<\/em>.<\/p>\r\n\r\n

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51.\u00a0<\/strong>[latex]\\left({x}^{2}+36\\right)\\frac{dy}{dx}=1,y\\left(6\\right)=0[\/latex]<\/div>\r\n
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\r\n\r\n52.\u00a0<\/strong>[latex]\\left(64-{x}^{2}\\right)\\frac{dy}{dx}=1,y\\left(0\\right)=3[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"614588\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"614588\"][latex]y=\\frac{1}{16}\\text{ln}|\\frac{x+8}{x - 8}|+3[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0<\/strong>Find the area bounded by [latex]y=\\frac{2}{\\sqrt{64 - 4{x}^{2}}},x=0,y=0,\\text{and }x=2[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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54.\u00a0<\/strong>An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\\frac{16}{\\sqrt{64+{x}^{2}}},x=0,y=0,x=2[\/latex] about the x<\/em>-axis. Find the volume of the tank (in cubic meters).<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"269654\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"269654\"]24.6 m3[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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55.\u00a0<\/strong>During each cycle, the velocity v<\/em> (in feet per second) of a robotic welding device is given by [latex]v=2t-\\frac{14}{4+{t}^{2}}[\/latex], where t<\/em> is time in seconds. Find the expression for the displacement s<\/em> (in feet) as a function of t<\/em> if [latex]s=0[\/latex] when [latex]t=0[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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56.\u00a0<\/strong>Find the length of the curve [latex]y=\\sqrt{16-{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=2[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"944365\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944365\"][latex]\\frac{2\\pi }{3}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Simplify the following expressions by writing each one using a single trigonometric function.<\/p>\n

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1.\u00a0<\/strong>[latex]4 - 4{\\sin}^{2}\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]9{\\sec}^{2}\\theta -9[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]9{\\tan}^{2}\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>[latex]{a}^{2}+{a}^{2}{\\tan}^{2}\\theta[\/latex]<\/div>\n
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4.\u00a0<\/strong>[latex]{a}^{2}+{a}^{2}{\\text{sinh}}^{2}\\theta[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]{a}^{2}{\\text{cosh}}^{2}\\theta[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]16{\\text{cosh}}^{2}\\theta -16[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use the technique of completing the square to express each trinomial as the square of a binomial.<\/p>\n

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6.\u00a0<\/strong>[latex]4{x}^{2}-4x+1[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
[latex]4{\\left(x-\\frac{1}{2}\\right)}^{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]2{x}^{2}-8x+3[\/latex]<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\text{-}{x}^{2}-2x+4[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{-}{\\left(x+1\\right)}^{2}+5[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.<\/span><\/p>\n<\/div>\n<\/div>\n

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9.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{4-{x}^{2}}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{ln}|x+\\sqrt{\\text{-}{a}^{2}+{x}^{2}}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{4-{x}^{2}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+9{x}^{2}}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{3}\\text{ln}|\\sqrt{9{x}^{2}+1}+3x|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
[latex]-\\frac{\\sqrt{1-{x}^{2}}}{x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left(1+{x}^{2}\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{2}+9}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]9\\left[\\frac{x\\sqrt{{x}^{2}+9}}{18}+\\frac{1}{2}ln|\\frac{\\sqrt{{x}^{2}+9}}{3}+\\frac{x}{3}|\\right]+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-25}}{x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{\\theta }^{3}d\\theta }{\\sqrt{9-{\\theta }^{2}}}d\\theta[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{1}{3}\\sqrt{9-{\\theta }^{2}}\\left(18+{\\theta }^{2}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
19.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{6}-{x}^{2}}}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{6}-{x}^{8}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{\\left(-1+{x}^{2}\\right)\\left(2+3{x}^{2}\\right)\\sqrt{{x}^{6}-{x}^{8}}}{15{x}^{3}}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left(1+{x}^{2}\\right)}^{\\frac{3}{2}}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{\\left({x}^{2}-9\\right)}^{\\frac{3}{2}}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{x}{9\\sqrt{-9+{x}^{2}}}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}dx}{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{{x}^{2}-1}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{2}\\left(\\text{ln}|x+\\sqrt{{x}^{2}-1}|+x\\sqrt{{x}^{2}-1}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{{x}^{2}+4}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{{x}^{2}+1}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{\\sqrt{1+{x}^{2}}}{x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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27.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}dx}{\\sqrt{1+{x}^{2}}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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28.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}{\\left(1-{x}^{2}\\right)}^{\\frac{3}{2}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{8}\\left(x\\left(5 - 2{x}^{2}\\right)\\sqrt{1-{x}^{2}}+3\\text{arcsin}x\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

In the following exercises, use the substitutions [latex]x=\\text{sinh}\\theta ,\\text{cosh}\\theta[\/latex], or [latex]\\text{tanh}\\theta[\/latex]. Express the final answers in terms of the variable <\/span>x.<\/em><\/p>\n<\/div>\n<\/div>\n

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29.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-1}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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30.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{ln}x-\\text{ln}|1+\\sqrt{1-{x}^{2}}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{{x}^{2}-1}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-1}}{{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{\\sqrt{-1+{x}^{2}}}{x}+\\text{ln}|x+\\sqrt{-1+{x}^{2}}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1-{x}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}}{{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n

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

Use the technique of completing the square to evaluate the following integrals.<\/p>\n

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35.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}-6x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{{x}^{2}+2x+1}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{1}{1+x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+2x+8}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+10x}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{2\\sqrt{-10+x}\\sqrt{x}\\text{ln}|\\sqrt{-10+x}+\\sqrt{x}|}{\\sqrt{\\left(10-x\\right)x}}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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39.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{{x}^{2}+4x - 12}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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40.\u00a0<\/strong>Evaluate the integral without using calculus: [latex]{\\displaystyle\\int }_{-3}^{3}\\sqrt{9-{x}^{2}}dx[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{9\\pi }{2}[\/latex]; area of a semicircle with radius 3<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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41.\u00a0<\/strong>Find the area enclosed by the ellipse [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex].<\/div>\n<\/div>\n<\/div>\n
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42.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex] using two different substitutions. First, let [latex]x=\\cos\\theta[\/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\\sin\\theta[\/latex] and use trigonometric substitution. Are the answers the same?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{arcsin}\\left(x\\right)+C[\/latex] is the common answer.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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43.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{{x}^{2}-1}}[\/latex] using the substitution [latex]x=\\sec\\theta[\/latex]. Next, evaluate the same integral using the substitution [latex]x=\\csc\\theta[\/latex]. Show that the results are equivalent.<\/div>\n<\/div>\n<\/div>\n
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44.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+1}dx[\/latex] using the form [latex]\\displaystyle\\int \\frac{1}{u}du[\/latex]. Next, evaluate the same integral using [latex]x=\\tan\\theta[\/latex]. Are the results the same?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex] is the result using either method.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int x\\sqrt{{x}^{2}+1}dx[\/latex]. Why did you choose this method?<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int {x}^{2}\\sqrt{{x}^{2}-1}dx[\/latex]. Why did you choose this method?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
Use trigonometric substitution. Let [latex]x=\\sec\\left(\\theta \\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{-1}^{1}\\frac{xdx}{{x}^{2}+1}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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48.\u00a0<\/strong>Find the length of the arc of the curve over the specified interval: [latex]y=\\text{ln}x,\\left[1,5\\right][\/latex]. Round the answer to three decimal places.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
4.367<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>Find the surface area of the solid generated by revolving the region bounded by the graphs of [latex]y={x}^{2},y=0,x=0,\\text{and }x=\\sqrt{2}[\/latex] about the x<\/em>-axis. (Round the answer to three decimal places).<\/div>\n<\/div>\n<\/div>\n
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50.\u00a0<\/strong>The region bounded by the graph of [latex]f\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex] and the x<\/em>-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex] is revolved about the x-<\/em>axis. Find the volume of the solid that is generated.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{{\\pi }^{2}}{8}+\\frac{\\pi }{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Solve the initial-value problem for y<\/em> as a function of x<\/em>.<\/p>\n

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51.\u00a0<\/strong>[latex]\\left({x}^{2}+36\\right)\\frac{dy}{dx}=1,y\\left(6\\right)=0[\/latex]<\/div>\n
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52.\u00a0<\/strong>[latex]\\left(64-{x}^{2}\\right)\\frac{dy}{dx}=1,y\\left(0\\right)=3[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]y=\\frac{1}{16}\\text{ln}|\\frac{x+8}{x - 8}|+3[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>Find the area bounded by [latex]y=\\frac{2}{\\sqrt{64 - 4{x}^{2}}},x=0,y=0,\\text{and }x=2[\/latex].<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\\frac{16}{\\sqrt{64+{x}^{2}}},x=0,y=0,x=2[\/latex] about the x<\/em>-axis. Find the volume of the tank (in cubic meters).<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
24.6 m3<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>During each cycle, the velocity v<\/em> (in feet per second) of a robotic welding device is given by [latex]v=2t-\\frac{14}{4+{t}^{2}}[\/latex], where t<\/em> is time in seconds. Find the expression for the displacement s<\/em> (in feet) as a function of t<\/em> if [latex]s=0[\/latex] when [latex]t=0[\/latex].<\/div>\n<\/div>\n<\/div>\n
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56.\u00a0<\/strong>Find the length of the curve [latex]y=\\sqrt{16-{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=2[\/latex].<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{2\\pi }{3}[\/latex]<\/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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