{"id":65,"date":"2021-03-25T02:20:49","date_gmt":"2021-03-25T02:20:49","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/partial-fractions-2\/"},"modified":"2021-11-17T02:38:07","modified_gmt":"2021-11-17T02:38:07","slug":"partial-fractions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/partial-fractions-2\/","title":{"raw":"Problem Set: Partial Fractions","rendered":"Problem Set: Partial Fractions"},"content":{"raw":"

Express the rational function as a sum or difference of two simpler rational expressions.<\/p>\r\n\r\n

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

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[reveal-answer q=\"431935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"431935\"][latex]-\\frac{2}{x+1}+\\frac{5}{2\\left(x+2\\right)}+\\frac{1}{2x}[\/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]\\frac{1}{{x}^{3}-x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>[latex]\\frac{3x+1}{{x}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"574934\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"574934\"][latex]\\frac{1}{{x}^{2}}+\\frac{3}{x}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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5.\u00a0<\/strong>[latex]\\frac{3{x}^{2}}{{x}^{2}+1}[\/latex] (Hint:<\/em> Use long division first.)<\/div>\r\n
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\r\n\r\n6.\u00a0<\/strong>[latex]\\frac{2{x}^{4}}{{x}^{2}-2x}[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"253756\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"253756\"][latex]2{x}^{2}+4x+8+\\frac{16}{x - 2}[\/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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7.\u00a0<\/strong>[latex]\\frac{1}{\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>[latex]\\frac{1}{{x}^{2}\\left(x - 1\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"639536\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"639536\"][latex]-\\frac{1}{{x}^{2}}-\\frac{1}{x}+\\frac{1}{x - 1}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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9.\u00a0<\/strong>[latex]\\frac{x}{{x}^{2}-4}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]\\frac{1}{x\\left(x - 1\\right)\\left(x - 2\\right)\\left(x - 3\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"827093\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"827093\"][latex]-\\frac{1}{2\\left(x - 2\\right)}+\\frac{1}{2\\left(x - 1\\right)}-\\frac{1}{6x}+\\frac{1}{6\\left(x - 3\\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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11.\u00a0<\/strong>[latex]\\frac{1}{{x}^{4}-1}=\\frac{1}{\\left(x+1\\right)\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]\\frac{3{x}^{2}}{{x}^{3}-1}=\\frac{3{x}^{2}}{\\left(x - 1\\right)\\left({x}^{2}+x+1\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"168207\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"168207\"][latex]\\frac{1}{x - 1}+\\frac{2x+1}{{x}^{2}+x+1}[\/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]\\frac{2x}{{\\left(x+2\\right)}^{2}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]\\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\\left(x+1\\right){\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"548204\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"548204\"][latex]\\frac{2}{x+1}+\\frac{x}{{x}^{2}+4}-\\frac{1}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the method of partial fractions to evaluate each of the following integrals.<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

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16.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x}{{x}^{2}+2x - 8}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"715448\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"715448\"][latex]\\text{-}\\text{ln}|2-x|+2\\text{ln}|4+x|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

17.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-x}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{{x}^{2}-4}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"428421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"428421\"][latex]\\frac{1}{2}\\text{ln}|4-{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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19.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\left(x - 1\\right)\\left(x - 2\\right)\\left(x - 3\\right)}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"481673\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"481673\"][latex]2\\left(x+\\frac{1}{3}\\text{arctan}\\left(\\frac{1+x}{3}\\right)\\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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21.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}-5x+6}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2-x}{{x}^{2}+x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"373859\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"373859\"][latex]2\\text{ln}|x|-3\\text{ln}|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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23.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2}{{x}^{2}-x - 6}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"884634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"884634\"][latex]\\frac{1}{16}\\left(\\text{-}\\frac{4}{-2+x}-\\text{ln}|-2+x|+\\text{ln}|2+x|\\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{dx}{{x}^{4}-10{x}^{2}+9}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Evaluate the following integrals, which have irreducible quadratic factors.<\/p>\r\n\r\n

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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2}{\\left(x - 4\\right)\\left({x}^{2}+2x+6\\right)}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"241304\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"241304\"][latex]\\frac{1}{30}\\left(-2\\sqrt{5}\\text{arctan}\\left[\\frac{1+x}{\\sqrt{5}}\\right]+2\\text{ln}|-4+x|-\\text{ln}|6+2x+{x}^{2}|\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

27.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}}{{x}^{3}-{x}^{2}+4x - 4}dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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28.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"131459\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"131459\"][latex]-\\frac{3}{x}+4\\text{ln}|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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29.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{\\left(x - 1\\right){\\left({x}^{2}+2x+2\\right)}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the method of partial fractions to evaluate the following integrals.<\/p>\r\n\r\n

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30.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x+4}{\\left({x}^{2}+4\\right)\\left(3-x\\right)}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"629370\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"629370\"][latex]\\text{-}\\text{ln}|3-x|+\\frac{1}{2}\\text{ln}|{x}^{2}+4|+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 \\frac{2}{{\\left(x+2\\right)}^{2}\\left(2-x\\right)}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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32.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x+4}{{x}^{3}-2x - 4}dx[\/latex] (Hint:<\/em> Use the rational root theorem.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"825994\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"825994\"][latex]\\text{ln}|x - 2|-\\frac{1}{2}\\text{ln}|{x}^{2}+2x+2|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.<\/p>\r\n\r\n

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33.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{{e}^{x}}{36-{e}^{2x}}dx[\/latex] (Give the exact answer and the decimal equivalent. Round to five decimal places.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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34.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"393315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"393315\"][latex]\\text{-}x+\\text{ln}|1-{e}^{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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35.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sin{x}dx}{1-{\\cos}^{2}x}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sin{x}}{{\\cos}^{2}x+\\cos{x} - 6}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"609077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"609077\"][latex]\\frac{1}{5}\\text{ln}|\\frac{\\cos{x}+3}{\\cos{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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37.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1-\\sqrt{x}}{1+\\sqrt{x}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dt}{{\\left({e}^{t}-{e}^{\\text{-}t}\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"447630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"447630\"][latex]\\frac{1}{2 - 2{e}^{2t}}+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+{e}^{x}}{1-{e}^{x}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1+\\sqrt{x+1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"61363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"61363\"][latex]2\\sqrt{1+x}-2\\text{ln}|1+\\sqrt{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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41.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{x}+\\sqrt[4]{x}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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42.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\cos{x}}{\\sin{x}\\left(1-\\sin{x}\\right)}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"30075\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"30075\"][latex]\\text{ln}|\\frac{\\sin{x}}{1-\\sin{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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43.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}}{{\\left({e}^{2x}-4\\right)}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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44.\u00a0<\/strong>[latex]\\underset{1}{\\overset{2}{\\displaystyle\\int }}\\frac{1}{{x}^{2}\\sqrt{4-{x}^{2}}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"50277\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"50277\"][latex]\\frac{\\sqrt{3}}{4}[\/latex][\/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>[latex]\\displaystyle\\int \\frac{1}{2+{e}^{\\text{-}x}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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46.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{1+{e}^{x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"300944\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"300944\"][latex]x-\\text{ln}\\left(1+{e}^{x}\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the given substitution to convert the integral to an integral of a rational function, then evaluate.<\/p>\r\n\r\n

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47.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{t-\\sqrt[3]{t}}dtt={x}^{3}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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48.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{x}+\\sqrt[3]{x}}dx;x={u}^{6}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"280083\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280083\"][latex]6{x}^{\\frac{1}{6}}-3{x}^{\\frac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left(1+{x}^{\\frac{1}{6}}\\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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49.\u00a0<\/strong>Graph the curve [latex]y=\\frac{x}{1+x}[\/latex] over the interval [latex]\\left[0,5\\right][\/latex]. Then, find the area of the region bounded by the curve, the x<\/em>-axis, and the line [latex]x=4[\/latex].<\/p>\r\n\r\n\"This<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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50.\u00a0<\/strong>Find the volume of the solid generated when the region bounded by [latex]y=\\frac{1}{\\sqrt{x\\left(3-x\\right)}}[\/latex], [latex]y=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex] is revolved about the x-<\/em>axis.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"195931\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"195931\"][latex]\\frac{4}{3}\\pi \\text{arctanh}\\left[\\frac{1}{3}\\right]=\\frac{1}{3}\\pi \\text{ln}4[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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51.\u00a0<\/strong>The velocity of a particle moving along a line is a function of time given by [latex]v\\left(t\\right)=\\frac{88{t}^{2}}{{t}^{2}+1}[\/latex]. Find the distance that the particle has traveled after [latex]t=5[\/latex] sec.<\/div>\r\n<\/div>\r\n<\/div>\r\n

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

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52.\u00a0<\/strong>[latex]\\left({t}^{2}-7t+12\\right)\\frac{dx}{dt}=1,\\left(t>4,x\\left(5\\right)=0\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"915\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915\"][latex]x=\\text{-}\\text{ln}|t - 3|+\\text{ln}|t - 4|+\\text{ln}2[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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53.\u00a0<\/strong>[latex]\\left(t+5\\right)\\frac{dx}{dt}={x}^{2}+1,t>\\text{-}5,x\\left(1\\right)=\\tan1[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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54.\u00a0<\/strong>[latex]\\left(2{t}^{3}-2{t}^{2}+t - 1\\right)\\frac{dx}{dt}=3,x\\left(2\\right)=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"524497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"524497\"][latex]x=\\text{ln}|t - 1|-\\sqrt{2}\\text{arctan}\\left(\\sqrt{2}t\\right)-\\frac{1}{2}\\text{ln}\\left({t}^{2}+\\frac{1}{2}\\right)+\\sqrt{2}\\text{arctan}\\left(2\\sqrt{2}\\right)+\\frac{1}{2}\\text{ln}4.5[\/latex][\/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>Find the x<\/em>-coordinate of the centroid of the area bounded by [latex]y\\left({x}^{2}-9\\right)=1[\/latex], [latex]y=0,x=4,\\text{and }x=5[\/latex]. (Round the answer to two decimal places.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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56.\u00a0<\/strong>Find the volume generated by revolving the area bounded by [latex]y=\\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\\text{and }y=0[\/latex] about the y<\/em>-axis.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"437705\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"437705\"][latex]\\frac{2}{5}\\pi \\text{ln}\\frac{28}{13}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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57.\u00a0<\/strong>Find the area bounded by [latex]y=\\frac{x - 12}{{x}^{2}-8x - 20}[\/latex], [latex]y=0,x=2,\\text{and }x=4[\/latex]. (Round the answer to the nearest hundredth.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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58.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}+1}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"13502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"13502\"][latex]\\frac{\\text{arctan}\\left[\\frac{-1+2x}{\\sqrt{3}}\\right]}{\\sqrt{3}}+\\frac{1}{3}\\text{ln}|1+x|-\\frac{1}{6}\\text{ln}|1-x+{x}^{2}|+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following problems, use the substitutions [latex]\\tan\\left(\\frac{x}{2}\\right)=t[\/latex], [latex]dx=\\frac{2}{1+{t}^{2}}dt[\/latex], [latex]\\sin{x}=\\frac{2t}{1+{t}^{2}}[\/latex], and [latex]\\cos{x}=\\frac{1-{t}^{2}}{1+{t}^{2}}[\/latex].<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{3 - 5\\sin{x}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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60.\u00a0<\/strong>Find the area under the curve [latex]y=\\frac{1}{1+\\sin{x}}[\/latex] between [latex]x=0[\/latex] and [latex]x=\\pi [\/latex]. (Assume the dimensions are in inches.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"906957\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"906957\"]2.0 in.2[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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61.\u00a0<\/strong>Given [latex]\\tan\\left(\\frac{x}{2}\\right)=t[\/latex], derive the formulas [latex]dx=\\frac{2}{1+{t}^{2}}dt[\/latex], [latex]\\sin{x}=\\frac{2t}{1+{t}^{2}}[\/latex], and [latex]\\cos{x}=\\frac{1-{t}^{2}}{1+{t}^{2}}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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62.\u00a0<\/strong>Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt[3]{x - 8}}{x}dx[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"325632\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"325632\"][latex]3{\\left(-8+x\\right)}^{\\frac{1}{3}}[\/latex] [latex]-2\\sqrt{3}\\text{arctan}\\left[\\frac{-1+{\\left(-8+x\\right)}^{\\frac{1}{3}}}{\\sqrt{3}}\\right][\/latex] [latex]-2\\text{ln}\\left[2+{\\left(-8+x\\right)}^{\\frac{1}{3}}\\right][\/latex] [latex]+\\text{ln}\\left[4 - 2{\\left(-8+x\\right)}^{\\frac{1}{3}}+{\\left(-8+x\\right)}^{\\frac{2}{3}}\\right]+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Express the rational function as a sum or difference of two simpler rational expressions.<\/p>\n

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1.<\/strong> [latex]\\frac{1}{\\left(x - 3\\right)\\left(x - 2\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]\\frac{{x}^{2}+1}{x\\left(x+1\\right)\\left(x+2\\right)}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{2}{x+1}+\\frac{5}{2\\left(x+2\\right)}+\\frac{1}{2x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>[latex]\\frac{1}{{x}^{3}-x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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4.\u00a0<\/strong>[latex]\\frac{3x+1}{{x}^{2}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{{x}^{2}}+\\frac{3}{x}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]\\frac{3{x}^{2}}{{x}^{2}+1}[\/latex] (Hint:<\/em> Use long division first.)<\/div>\n
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6.\u00a0<\/strong>[latex]\\frac{2{x}^{4}}{{x}^{2}-2x}[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]2{x}^{2}+4x+8+\\frac{16}{x - 2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]\\frac{1}{\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\frac{1}{{x}^{2}\\left(x - 1\\right)}[\/latex]<\/p>\n<\/div>\n

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

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Show Solution<\/span><\/p>\n
[latex]-\\frac{1}{2\\left(x - 2\\right)}+\\frac{1}{2\\left(x - 1\\right)}-\\frac{1}{6x}+\\frac{1}{6\\left(x - 3\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>[latex]\\frac{1}{{x}^{4}-1}=\\frac{1}{\\left(x+1\\right)\\left(x - 1\\right)\\left({x}^{2}+1\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]\\frac{3{x}^{2}}{{x}^{3}-1}=\\frac{3{x}^{2}}{\\left(x - 1\\right)\\left({x}^{2}+x+1\\right)}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{x - 1}+\\frac{2x+1}{{x}^{2}+x+1}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]\\frac{2x}{{\\left(x+2\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]\\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\\left(x+1\\right){\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{2}{x+1}+\\frac{x}{{x}^{2}+4}-\\frac{1}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Use the method of partial fractions to evaluate each of the following integrals.<\/span><\/p>\n<\/div>\n<\/div>\n

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15.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\left(x - 3\\right)\\left(x - 2\\right)}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x}{{x}^{2}+2x - 8}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{-}\\text{ln}|2-x|+2\\text{ln}|4+x|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
17.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-x}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{{x}^{2}-4}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{2}\\text{ln}|4-{x}^{2}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{x\\left(x - 1\\right)\\left(x - 2\\right)\\left(x - 3\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]2\\left(x+\\frac{1}{3}\\text{arctan}\\left(\\frac{1+x}{3}\\right)\\right)+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}{{x}^{2}-5x+6}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2-x}{{x}^{2}+x}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]2\\text{ln}|x|-3\\text{ln}|1+x|+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{2}{{x}^{2}-x - 6}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{16}\\left(\\text{-}\\frac{4}{-2+x}-\\text{ln}|-2+x|+\\text{ln}|2+x|\\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{dx}{{x}^{4}-10{x}^{2}+9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Evaluate the following integrals, which have irreducible quadratic factors.<\/p>\n

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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2}{\\left(x - 4\\right)\\left({x}^{2}+2x+6\\right)}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{30}\\left(-2\\sqrt{5}\\text{arctan}\\left[\\frac{1+x}{\\sqrt{5}}\\right]+2\\text{ln}|-4+x|-\\text{ln}|6+2x+{x}^{2}|\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
27.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{2}}{{x}^{3}-{x}^{2}+4x - 4}dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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28.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]-\\frac{3}{x}+4\\text{ln}|x+2|+x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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29.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{\\left(x - 1\\right){\\left({x}^{2}+2x+2\\right)}^{2}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use the method of partial fractions to evaluate the following integrals.<\/p>\n

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30.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x+4}{\\left({x}^{2}+4\\right)\\left(3-x\\right)}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{-}\\text{ln}|3-x|+\\frac{1}{2}\\text{ln}|{x}^{2}+4|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{2}{{\\left(x+2\\right)}^{2}\\left(2-x\\right)}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x+4}{{x}^{3}-2x - 4}dx[\/latex] (Hint:<\/em> Use the rational root theorem.)<\/p>\n<\/div>\n

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

Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.<\/p>\n

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33.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{{e}^{x}}{36-{e}^{2x}}dx[\/latex] (Give the exact answer and the decimal equivalent. Round to five decimal places.)<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\text{-}x+\\text{ln}|1-{e}^{x}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sin{x}dx}{1-{\\cos}^{2}x}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\sin{x}}{{\\cos}^{2}x+\\cos{x} - 6}dx[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{1}{5}\\text{ln}|\\frac{\\cos{x}+3}{\\cos{x} - 2}|+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{x}}{1+\\sqrt{x}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dt}{{\\left({e}^{t}-{e}^{\\text{-}t}\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n

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

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

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

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Show Solution<\/span><\/p>\n
[latex]\\frac{\\sqrt{3}}{4}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{2+{e}^{\\text{-}x}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{1+{e}^{x}}dx[\/latex]<\/p>\n<\/div>\n

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

Use the given substitution to convert the integral to an integral of a rational function, then evaluate.<\/p>\n

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47.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{t-\\sqrt[3]{t}}dtt={x}^{3}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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48.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{x}+\\sqrt[3]{x}}dx;x={u}^{6}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]6{x}^{\\frac{1}{6}}-3{x}^{\\frac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left(1+{x}^{\\frac{1}{6}}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>Graph the curve [latex]y=\\frac{x}{1+x}[\/latex] over the interval [latex]\\left[0,5\\right][\/latex]. Then, find the area of the region bounded by the curve, the x<\/em>-axis, and the line [latex]x=4[\/latex].<\/p>\n


\n\"This<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n

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50.\u00a0<\/strong>Find the volume of the solid generated when the region bounded by [latex]y=\\frac{1}{\\sqrt{x\\left(3-x\\right)}}[\/latex], [latex]y=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex] is revolved about the x-<\/em>axis.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
[latex]\\frac{4}{3}\\pi \\text{arctanh}\\left[\\frac{1}{3}\\right]=\\frac{1}{3}\\pi \\text{ln}4[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>The velocity of a particle moving along a line is a function of time given by [latex]v\\left(t\\right)=\\frac{88{t}^{2}}{{t}^{2}+1}[\/latex]. Find the distance that the particle has traveled after [latex]t=5[\/latex] sec.<\/div>\n<\/div>\n<\/div>\n

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

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52.\u00a0<\/strong>[latex]\\left({t}^{2}-7t+12\\right)\\frac{dx}{dt}=1,\\left(t>4,x\\left(5\\right)=0\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]x=\\text{-}\\text{ln}|t - 3|+\\text{ln}|t - 4|+\\text{ln}2[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>[latex]\\left(t+5\\right)\\frac{dx}{dt}={x}^{2}+1,t>\\text{-}5,x\\left(1\\right)=\\tan1[\/latex]<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>[latex]\\left(2{t}^{3}-2{t}^{2}+t - 1\\right)\\frac{dx}{dt}=3,x\\left(2\\right)=0[\/latex]<\/p>\n<\/div>\n

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[latex]x=\\text{ln}|t - 1|-\\sqrt{2}\\text{arctan}\\left(\\sqrt{2}t\\right)-\\frac{1}{2}\\text{ln}\\left({t}^{2}+\\frac{1}{2}\\right)+\\sqrt{2}\\text{arctan}\\left(2\\sqrt{2}\\right)+\\frac{1}{2}\\text{ln}4.5[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>Find the x<\/em>-coordinate of the centroid of the area bounded by [latex]y\\left({x}^{2}-9\\right)=1[\/latex], [latex]y=0,x=4,\\text{and }x=5[\/latex]. (Round the answer to two decimal places.)<\/div>\n<\/div>\n<\/div>\n
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56.\u00a0<\/strong>Find the volume generated by revolving the area bounded by [latex]y=\\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\\text{and }y=0[\/latex] about the y<\/em>-axis.<\/p>\n<\/div>\n

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[latex]\\frac{2}{5}\\pi \\text{ln}\\frac{28}{13}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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57.\u00a0<\/strong>Find the area bounded by [latex]y=\\frac{x - 12}{{x}^{2}-8x - 20}[\/latex], [latex]y=0,x=2,\\text{and }x=4[\/latex]. (Round the answer to the nearest hundredth.)<\/div>\n<\/div>\n<\/div>\n
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58.\u00a0<\/strong>Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}+1}[\/latex].<\/p>\n<\/div>\n

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[latex]\\frac{\\text{arctan}\\left[\\frac{-1+2x}{\\sqrt{3}}\\right]}{\\sqrt{3}}+\\frac{1}{3}\\text{ln}|1+x|-\\frac{1}{6}\\text{ln}|1-x+{x}^{2}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following problems, use the substitutions [latex]\\tan\\left(\\frac{x}{2}\\right)=t[\/latex], [latex]dx=\\frac{2}{1+{t}^{2}}dt[\/latex], [latex]\\sin{x}=\\frac{2t}{1+{t}^{2}}[\/latex], and [latex]\\cos{x}=\\frac{1-{t}^{2}}{1+{t}^{2}}[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n

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59.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{3 - 5\\sin{x}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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60.\u00a0<\/strong>Find the area under the curve [latex]y=\\frac{1}{1+\\sin{x}}[\/latex] between [latex]x=0[\/latex] and [latex]x=\\pi[\/latex]. (Assume the dimensions are in inches.)<\/p>\n<\/div>\n

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2.0 in.2<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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61.\u00a0<\/strong>Given [latex]\\tan\\left(\\frac{x}{2}\\right)=t[\/latex], derive the formulas [latex]dx=\\frac{2}{1+{t}^{2}}dt[\/latex], [latex]\\sin{x}=\\frac{2t}{1+{t}^{2}}[\/latex], and [latex]\\cos{x}=\\frac{1-{t}^{2}}{1+{t}^{2}}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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62.\u00a0<\/strong>Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt[3]{x - 8}}{x}dx[\/latex].<\/p>\n<\/div>\n

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[latex]3{\\left(-8+x\\right)}^{\\frac{1}{3}}[\/latex] [latex]-2\\sqrt{3}\\text{arctan}\\left[\\frac{-1+{\\left(-8+x\\right)}^{\\frac{1}{3}}}{\\sqrt{3}}\\right][\/latex] [latex]-2\\text{ln}\\left[2+{\\left(-8+x\\right)}^{\\frac{1}{3}}\\right][\/latex] [latex]+\\text{ln}\\left[4 - 2{\\left(-8+x\\right)}^{\\frac{1}{3}}+{\\left(-8+x\\right)}^{\\frac{2}{3}}\\right]+C[\/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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