{"id":82,"date":"2021-03-25T02:20:54","date_gmt":"2021-03-25T02:20:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/other-strategies-for-integration-3\/"},"modified":"2021-11-17T02:38:24","modified_gmt":"2021-11-17T02:38:24","slug":"other-strategies-for-integration-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/other-strategies-for-integration-3\/","title":{"raw":"Problem Set: Other Strategies for Integration","rendered":"Problem Set: Other Strategies for Integration"},"content":{"raw":"

Use a table of integrals to evaluate the following integrals.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]\\underset{0}{\\overset{4}{\\displaystyle\\int }}\\frac{x}{\\sqrt{1+2x}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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2.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x+3}{{x}^{2}+2x+2}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"555280\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"555280\"][latex]\\frac{1}{2}\\text{ln}|{x}^{2}+2x+2|+2\\text{arctan}\\left(x+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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3.\u00a0<\/strong>[latex]\\displaystyle\\int {x}^{3}\\sqrt{1+2{x}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{\\sqrt{{x}^{2}+6x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"278339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278339\"][latex]{\\text{cosh}}^{-1}\\left(\\frac{x+3}{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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5.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x}{x+1}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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6.\u00a0<\/strong>[latex]\\displaystyle\\int x\\cdot {2}^{{x}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"475106\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"475106\"][latex]\\frac{{2}^{{x}^{2}-1}}{\\text{ln}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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7.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{4{x}^{2}+25}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dy}{\\sqrt{4-{y}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"223418\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"223418\"][latex]\\text{arcsin}\\left(\\frac{y}{2}\\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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9.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}\\left(2x\\right)\\cos\\left(2x\\right)dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\csc\\left(2w\\right)\\cot\\left(2w\\right)dw[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"51959\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"51959\"][latex]-\\frac{1}{2}\\csc\\left(2w\\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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11.\u00a0<\/strong>[latex]\\displaystyle\\int {2}^{y}dy[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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12.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{3xdx}{\\sqrt{{x}^{2}+8}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"359640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359640\"][latex]9 - 6\\sqrt{2}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{-1}{4}}^{\\frac{1}{4}}{\\sec}^{2}\\left(\\pi x\\right)\\tan\\left(\\pi x\\right)dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\tan}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"161000\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161000\"][latex]2-\\frac{\\pi }{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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15.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{3}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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16. <\/strong>[latex]\\displaystyle\\int {\\tan}^{5}\\left(3x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"173833\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"173833\"][latex]\\frac{1}{12}{\\tan}^{4}\\left(3x\\right)-\\frac{1}{6}{\\tan}^{2}\\left(3x\\right)+\\frac{1}{3}\\text{ln}|\\sec\\left(3x\\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 {\\sin}^{2}y{\\cos}^{3}ydy[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.<\/p>\r\n\r\n

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18.<\/strong> [T]<\/strong> [latex]\\displaystyle\\int \\frac{dw}{1+\\sec\\left(\\frac{w}{2}\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"38823\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"38823\"][latex]2\\cot\\left(\\frac{w}{2}\\right)-2\\csc\\left(\\frac{w}{2}\\right)+w+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.<\/strong> [T]<\/strong> [latex]\\displaystyle\\int \\frac{dw}{1-\\cos\\left(7w\\right)}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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20.\u00a0[T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{t}\\frac{dt}{4\\cos{t}+3\\sin{t}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"203961\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"203961\"][latex]\\frac{1}{5}\\text{ln}|\\frac{2\\left(5+4\\sin{t} - 3\\cos{t}\\right)}{4\\cos{t}+3\\sin{t}}|[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

21. [T]<\/strong> [latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-9}}{3x}dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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22. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{{x}^{\\frac{1}{2}}+{x}^{\\frac{1}{3}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"624438\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"624438\"][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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23. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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24. [T]<\/strong> [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"47349\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"47349\"][latex]\\text{-}{x}^{3}\\cos{x}+3{x}^{2}\\sin{x}+6x\\cos{x} - 6\\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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25. [T]<\/strong> [latex]\\displaystyle\\int x\\sqrt{{x}^{4}-9}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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26. [T]<\/strong> [latex]\\displaystyle\\int \\frac{x}{1+{e}^{\\text{-}{x}^{2}}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"11499\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"11499\"][latex]\\frac{1}{2}\\left({x}^{2}+\\text{ln}|1+{e}^{\\text{-}{x}^{2}}|\\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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27. [T]<\/strong> [latex]\\displaystyle\\int \\frac{\\sqrt{3 - 5x}}{2x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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28. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"560821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"560821\"][latex]2\\text{arctan}\\left(\\sqrt{x - 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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29. [T]<\/strong> [latex]\\displaystyle\\int {e}^{x}{\\cos}^{-1}\\left({e}^{x}\\right)dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use a calculator or CAS to evaluate the following integrals.<\/p>\r\n\r\n

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30. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\frac{z\\pi}{4}}\\cos\\left(2x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"184267\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"184267\"][latex]0.5=\\frac{1}{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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31. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{1}x\\cdot {e}^{\\text{-}{x}^{2}}dx[\/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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32. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{8}\\frac{2x}{\\sqrt{{x}^{2}+36}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"821851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"821851\"]8.0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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33. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\frac{2}{\\sqrt{3}}}\\frac{1}{4+9{x}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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34. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+4x+13}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"384139\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"384139\"][latex]\\frac{1}{3}\\text{arctan}\\left(\\frac{1}{3}\\left(x+2\\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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35. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{1+\\sin{x}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.<\/p>\r\n\r\n

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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+2x+10}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"800480\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"800480\"][latex]\\frac{1}{3}\\text{arctan}\\left(\\frac{x+1}{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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37.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-6x}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-4}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"296858\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"296858\"][latex]\\text{ln}\\left({e}^{x}+\\sqrt{4+{e}^{2x}}\\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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39.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\cos{x}}{{\\sin}^{2}x+2\\sin{x}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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40.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{arctan}\\left({x}^{3}\\right)}{{x}^{4}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"907598\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"907598\"][latex]\\text{ln}x-\\frac{1}{6}\\text{ln}\\left({x}^{6}+1\\right)-\\frac{\\text{arctan}\\left({x}^{3}\\right)}{3{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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41.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{ln}|x|\\text{arcsin}\\left(\\text{ln}|x|\\right)}{x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use tables to perform the integration.<\/p>\r\n\r\n

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

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[reveal-answer q=\"895240\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"895240\"][latex]\\text{ln}|x+\\sqrt{16+{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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43.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{3x}{2x+7}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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44.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1-\\cos\\left(4x\\right)}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"192477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"192477\"][latex]-\\frac{1}{4}\\cot\\left(2x\\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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45.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{4x+1}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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46.\u00a0<\/strong>Find the area bounded by [latex]y\\left(4+25{x}^{2}\\right)=5,x=0,y=0,\\text{and }x=4[\/latex]. Use a table of integrals or a CAS.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"656813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"656813\"][latex]\\frac{1}{2}\\text{arctan}10[\/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>The region bounded between the curve [latex]y=\\frac{1}{\\sqrt{1+\\cos{x}}},0.3\\le x\\le 1.1[\/latex], and the x<\/em>-axis is revolved about the x<\/em>-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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48.\u00a0<\/strong>Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\\le x\\le 3[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"608856\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"608856\"]1276.14[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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49. [T]<\/strong> Use an integral table and a calculator to find the area of the surface generated by revolving the curve [latex]y=\\frac{{x}^{2}}{2},0\\le x\\le 1[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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50. [T]<\/strong> Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\\cos{x},0\\le x\\le \\frac{\\pi }{2}[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"158220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"158220\"]7.21[\/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>Find the length of the curve [latex]y=\\frac{{x}^{2}}{4}[\/latex] over [latex]\\left[0,8\\right][\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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52.\u00a0<\/strong>Find the length of the curve [latex]y={e}^{x}[\/latex] over [latex]\\left[0,\\text{ln}\\left(2\\right)\\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"259373\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"259373\"][latex]\\sqrt{5}-\\sqrt{2}+\\text{ln}|\\frac{2+2\\sqrt{2}}{1+\\sqrt{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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53.\u00a0<\/strong>Find the area of the surface formed by revolving the graph of [latex]y=2\\sqrt{x}[\/latex] over the interval [latex]\\left[0,9\\right][\/latex] about the x<\/em>-axis.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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54.\u00a0<\/strong>Find the average value of the function [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}+1}[\/latex] over the interval [latex]\\left[-3,3\\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"84539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"84539\"][latex]\\frac{1}{3}\\text{arctan}\\left(3\\right)\\approx 0.416[\/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>Approximate the arc length of the curve [latex]y=\\tan\\left(\\pi x\\right)[\/latex] over the interval [latex]\\left[0,\\frac{1}{4}\\right][\/latex]. (Round the answer to three decimal places.)<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Use a table of integrals to evaluate the following integrals.<\/p>\n

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1.\u00a0<\/strong>[latex]\\underset{0}{\\overset{4}{\\displaystyle\\int }}\\frac{x}{\\sqrt{1+2x}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{x+3}{{x}^{2}+2x+2}dx[\/latex]<\/p>\n<\/div>\n

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

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

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[latex]\\frac{{2}^{{x}^{2}-1}}{\\text{ln}2}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{1}{4{x}^{2}+25}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dy}{\\sqrt{4-{y}^{2}}}[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{arcsin}\\left(\\frac{y}{2}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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9.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}\\left(2x\\right)\\cos\\left(2x\\right)dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]\\displaystyle\\int \\csc\\left(2w\\right)\\cot\\left(2w\\right)dw[\/latex]<\/p>\n<\/div>\n

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[latex]-\\frac{1}{2}\\csc\\left(2w\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>[latex]\\displaystyle\\int {2}^{y}dy[\/latex]<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{3xdx}{\\sqrt{{x}^{2}+8}}[\/latex]<\/p>\n<\/div>\n

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[latex]9 - 6\\sqrt{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{-1}{4}}^{\\frac{1}{4}}{\\sec}^{2}\\left(\\pi x\\right)\\tan\\left(\\pi x\\right)dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\tan}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/p>\n<\/div>\n

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[latex]2-\\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{3}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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16. <\/strong>[latex]\\displaystyle\\int {\\tan}^{5}\\left(3x\\right)dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{12}{\\tan}^{4}\\left(3x\\right)-\\frac{1}{6}{\\tan}^{2}\\left(3x\\right)+\\frac{1}{3}\\text{ln}|\\sec\\left(3x\\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 {\\sin}^{2}y{\\cos}^{3}ydy[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use a CAS to evaluate the following integrals. Tables can also be used to verify the answers.<\/p>\n

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18.<\/strong> [T]<\/strong> [latex]\\displaystyle\\int \\frac{dw}{1+\\sec\\left(\\frac{w}{2}\\right)}[\/latex]<\/p>\n<\/div>\n

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[latex]2\\cot\\left(\\frac{w}{2}\\right)-2\\csc\\left(\\frac{w}{2}\\right)+w+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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19.<\/strong> [T]<\/strong> [latex]\\displaystyle\\int \\frac{dw}{1-\\cos\\left(7w\\right)}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0[T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{t}\\frac{dt}{4\\cos{t}+3\\sin{t}}[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{5}\\text{ln}|\\frac{2\\left(5+4\\sin{t} - 3\\cos{t}\\right)}{4\\cos{t}+3\\sin{t}}|[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
21. [T]<\/strong> [latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-9}}{3x}dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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22. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{{x}^{\\frac{1}{2}}+{x}^{\\frac{1}{3}}}[\/latex]<\/p>\n<\/div>\n

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[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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23. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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24. [T]<\/strong> [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{-}{x}^{3}\\cos{x}+3{x}^{2}\\sin{x}+6x\\cos{x} - 6\\sin{x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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25. [T]<\/strong> [latex]\\displaystyle\\int x\\sqrt{{x}^{4}-9}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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26. [T]<\/strong> [latex]\\displaystyle\\int \\frac{x}{1+{e}^{\\text{-}{x}^{2}}}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{2}\\left({x}^{2}+\\text{ln}|1+{e}^{\\text{-}{x}^{2}}|\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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27. [T]<\/strong> [latex]\\displaystyle\\int \\frac{\\sqrt{3 - 5x}}{2x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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28. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/p>\n<\/div>\n

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[latex]2\\text{arctan}\\left(\\sqrt{x - 1}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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29. [T]<\/strong> [latex]\\displaystyle\\int {e}^{x}{\\cos}^{-1}\\left({e}^{x}\\right)dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use a calculator or CAS to evaluate the following integrals.<\/p>\n

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30. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\frac{z\\pi}{4}}\\cos\\left(2x\\right)dx[\/latex]<\/p>\n<\/div>\n

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[latex]0.5=\\frac{1}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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31. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{1}x\\cdot {e}^{\\text{-}{x}^{2}}dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{8}\\frac{2x}{\\sqrt{{x}^{2}+36}}dx[\/latex]<\/p>\n<\/div>\n

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8.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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33. [T]<\/strong> [latex]{\\displaystyle\\int }_{0}^{\\frac{2}{\\sqrt{3}}}\\frac{1}{4+9{x}^{2}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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34. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+4x+13}[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{3}\\text{arctan}\\left(\\frac{1}{3}\\left(x+2\\right)\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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35. [T]<\/strong> [latex]\\displaystyle\\int \\frac{dx}{1+\\sin{x}}[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table.<\/p>\n

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36.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+2x+10}[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{3}\\text{arctan}\\left(\\frac{x+1}{3}\\right)+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{dx}{\\sqrt{{x}^{2}-6x}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-4}}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ln}\\left({e}^{x}+\\sqrt{4+{e}^{2x}}\\right)+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{\\cos{x}}{{\\sin}^{2}x+2\\sin{x}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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40.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{\\text{arctan}\\left({x}^{3}\\right)}{{x}^{4}}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{ln}x-\\frac{1}{6}\\text{ln}\\left({x}^{6}+1\\right)-\\frac{\\text{arctan}\\left({x}^{3}\\right)}{3{x}^{3}}+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{\\text{ln}|x|\\text{arcsin}\\left(\\text{ln}|x|\\right)}{x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use tables to perform the integration.<\/p>\n

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

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[latex]\\text{ln}|x+\\sqrt{16+{x}^{2}}|+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{3x}{2x+7}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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44.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{1-\\cos\\left(4x\\right)}[\/latex]<\/p>\n<\/div>\n

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[latex]-\\frac{1}{4}\\cot\\left(2x\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{dx}{\\sqrt{4x+1}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>Find the area bounded by [latex]y\\left(4+25{x}^{2}\\right)=5,x=0,y=0,\\text{and }x=4[\/latex]. Use a table of integrals or a CAS.<\/p>\n<\/div>\n

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[latex]\\frac{1}{2}\\text{arctan}10[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>The region bounded between the curve [latex]y=\\frac{1}{\\sqrt{1+\\cos{x}}},0.3\\le x\\le 1.1[\/latex], and the x<\/em>-axis is revolved about the x<\/em>-axis to generate a solid. Use a table of integrals to find the volume of the solid generated. (Round the answer to two decimal places.)<\/div>\n<\/div>\n<\/div>\n
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48.\u00a0<\/strong>Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\\le x\\le 3[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/p>\n<\/div>\n

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1276.14<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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49. [T]<\/strong> Use an integral table and a calculator to find the area of the surface generated by revolving the curve [latex]y=\\frac{{x}^{2}}{2},0\\le x\\le 1[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/div>\n<\/div>\n<\/div>\n
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50. [T]<\/strong> Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\\cos{x},0\\le x\\le \\frac{\\pi }{2}[\/latex], about the x<\/em>-axis. (Round the answer to two decimal places.)<\/p>\n<\/div>\n

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7.21<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>Find the length of the curve [latex]y=\\frac{{x}^{2}}{4}[\/latex] over [latex]\\left[0,8\\right][\/latex].<\/div>\n<\/div>\n<\/div>\n
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52.\u00a0<\/strong>Find the length of the curve [latex]y={e}^{x}[\/latex] over [latex]\\left[0,\\text{ln}\\left(2\\right)\\right][\/latex].<\/p>\n<\/div>\n

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[latex]\\sqrt{5}-\\sqrt{2}+\\text{ln}|\\frac{2+2\\sqrt{2}}{1+\\sqrt{5}}|[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>Find the area of the surface formed by revolving the graph of [latex]y=2\\sqrt{x}[\/latex] over the interval [latex]\\left[0,9\\right][\/latex] about the x<\/em>-axis.<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>Find the average value of the function [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}+1}[\/latex] over the interval [latex]\\left[-3,3\\right][\/latex].<\/p>\n<\/div>\n

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[latex]\\frac{1}{3}\\text{arctan}\\left(3\\right)\\approx 0.416[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>Approximate the arc length of the curve [latex]y=\\tan\\left(\\pi x\\right)[\/latex] over the interval [latex]\\left[0,\\frac{1}{4}\\right][\/latex]. (Round the answer to three decimal places.)<\/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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