{"id":77,"date":"2021-03-25T02:20:53","date_gmt":"2021-03-25T02:20:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-3\/"},"modified":"2022-01-03T18:26:47","modified_gmt":"2022-01-03T18:26:47","slug":"trigonometric-integrals-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/trigonometric-integrals-3\/","title":{"raw":"Problem Set: Trigonometric Integrals","rendered":"Problem Set: Trigonometric Integrals"},"content":{"raw":"

Fill in the blank to make a true statement.<\/p>\r\n\r\n

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1.\u00a0<\/strong>[latex]{\\sin}^{2}x+\\_\\_\\_\\_\\_\\_\\_=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"161391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161391\"][latex]{\\cos}^{2}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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2.\u00a0<\/strong>[latex]{\\sec}^{2}x - 1=\\_\\_\\_\\_\\_\\_\\_[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.<\/p>\r\n\r\n

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\r\n\r\n3.\u00a0<\/strong>[latex]{\\sin}^{2}x=\\_\\_\\_\\_\\_\\_\\_[\/latex]\r\n\r\n[reveal-answer q=\"453971\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453971\"][latex]\\frac{1-\\cos\\left(2x\\right)}{2}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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4.\u00a0<\/strong>[latex]{\\cos}^{2}x=\\_\\_\\_\\_\\_\\_\\_[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Evaluate each of the following integrals by u<\/em>-substitution.<\/p>\r\n\r\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}x\\cos{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"813484\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"813484\"][latex]\\frac{{\\sin}^{4}x}{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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6.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\cos{x}}\\sin{x}dx[\/latex]<\/div>\r\n
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\r\n\r\n7.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{5}\\left(2x\\right){\\sec}^{2}\\left(2x\\right)dx[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"866631\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"866631\"][latex]\\frac{1}{12}{\\tan}^{6}\\left(2x\\right)+C[\/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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8.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{7}\\left(2x\\right)\\cos\\left(2x\\right)dx[\/latex]<\/div>\r\n
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\r\n\r\n9.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan\\left(\\frac{x}{2}\\right){\\sec}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"957882\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"957882\"][latex]{\\sec}^{2}\\left(\\frac{x}{2}\\right)+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

10.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{2}x{\\sec}^{2}xdx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note<\/em>: Some of the problems may be done using techniques of integration learned previously.)<\/p>\r\n\r\n

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11.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"62893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"62893\"][latex]\\text{-}\\cos{x}+\\frac{1}{3}\\cos^{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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12.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{3}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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13.\u00a0<\/strong>[latex]\\displaystyle\\int \\sin{x}\\cos{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"795672\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"795672\"][latex]-\\frac{1}{2}{\\cos}^{2}x+C[\/latex] or [latex]\\frac{1}{2}{\\sin}^{2}x+C[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

14.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{5}xdx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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15.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{5}x{\\cos}^{2}xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"951500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951500\"][latex]-\\frac{1}{3}\\cos^{3}x+\\frac{2}{5}\\cos^{5}x-\\frac{1}{7}\\cos^{7}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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16. <\/strong>[latex]\\displaystyle\\int {\\sin}^{3}x{\\cos}^{3}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\sin{x}}\\cos{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"698490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698490\"][latex]\\frac{2}{3}{\\left(\\sin{x}\\right)}^{\\frac{3}{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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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\sin{x}}{\\cos}^{3}xdx[\/latex]<\/div>\r\n
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\r\n\r\n19.\u00a0<\/strong>[latex]\\displaystyle\\int \\sec{x}\\tan{x}dx[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"367672\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"367672\"][latex]\\sec{x}+C[\/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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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan\\left(5x\\right)dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{2}x\\sec{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"368969\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368969\"][latex]\\frac{1}{2}\\sec{x}\\tan{x}-\\frac{1}{2}\\text{ln}\\left(\\sec{x}+\\tan{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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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan{x}{\\sec}^{3}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sec}^{4}xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"867747\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867747\"][latex]\\frac{2\\tan{x}}{3}+\\frac{1}{3}\\sec{\\left(x\\right)}^{2}\\tan{x}[\/latex] [latex]=\\tan{x}+\\frac{{\\tan}^{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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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\cot{x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\csc{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"772297\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"772297\"][latex]\\text{-}\\text{ln}|\\cot{x}+\\csc{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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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{\\tan}^{3}x}{\\sqrt{\\sec{x}}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, find a general formula for the integrals.<\/p>\r\n\r\n

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27.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}ax\\cos{ax} dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"854173\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"854173\"][latex]\\frac{{\\sin}^{3}\\left(ax\\right)}{3a}+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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28. <\/strong>[latex]\\displaystyle\\int \\sin{ax} \\cos{ax} dx[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n

Use the double-angle formulas to evaluate the following integrals.<\/p>\r\n\r\n

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29.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"556856\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556856\"][latex]\\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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30.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{4}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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31.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{2}3xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"452749\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"452749\"][latex]\\frac{x}{2}+\\frac{1}{12}\\sin\\left(6x\\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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32.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}xdx[\/latex]<\/div>\r\n
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\r\n\r\n33.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}xdx+\\displaystyle\\int {\\cos}^{2}xdx[\/latex]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"594007\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"594007\"][latex]x+C[\/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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34.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}\\left(2x\\right)dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.<\/p>\r\n\r\n

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35.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2\\pi }\\cos{x}\\sin2xdx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"960701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"960701\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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36.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }\\sin3x\\sin5xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }\\cos\\left(99x\\right)\\sin\\left(101x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"913235\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"913235\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

38.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }{\\cos}^{2}\\left(3x\\right)dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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39.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sin{x}\\sin\\left(2x\\right)\\sin\\left(3x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"273497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"273497\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

40.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4\\pi }\\cos\\left(\\frac{x}{2}\\right)\\sin\\left(\\frac{x}{2}\\right)dx[\/latex]<\/span><\/div>\r\n
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\r\n\r\n41.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{\\pi}{6}}^{\\frac{\\pi}{3}}\\frac{{\\cos}^{3}x}{\\sqrt{\\sin{x}}}dx[\/latex] (Round this answer to three decimal places.)<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"176711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176711\"]Approximately 0.239[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

42.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{\\text{-}\\pi}{3}}^{\\frac{\\pi}{3}}\\sqrt{{\\sec}^{2}x - 1}dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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43.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-\\cos\\left(2x\\right)}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"998634\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"998634\"][latex]\\sqrt{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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44.\u00a0<\/strong>Find the area of the region bounded by the graphs of the equations [latex]y=\\sin{x},y={\\sin}^{3}x,x=0,\\text{and }x=\\frac{\\pi }{2}[\/latex].<\/div>\r\n
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\r\n\r\n45.\u00a0<\/strong>Find the area of the region bounded by the graphs of the equations [latex]y={\\cos}^{2}x,y={\\sin}^{2}x,x=-\\frac{\\pi }{4},\\text{and }x=\\frac{\\pi }{4}[\/latex].<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n
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[reveal-answer q=\"362874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"362874\"]1.0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

46.\u00a0<\/strong>A particle moves in a straight line with the velocity function [latex]v\\left(t\\right)=\\sin\\left(\\omega t\\right){\\cos}^{2}\\left(\\omega t\\right)[\/latex]. Find its position function [latex]x=f\\left(t\\right)[\/latex] if [latex]f\\left(0\\right)=0[\/latex].<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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47.\u00a0<\/strong>Find the average value of the function [latex]f\\left(x\\right)={\\sin}^{2}x{\\cos}^{3}x[\/latex] over the interval [latex]\\left[\\text{-}\\pi ,\\pi \\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"673059\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"673059\"]0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, solve the differential equations.<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>[latex]\\frac{dy}{dx}={\\sin}^{2}x[\/latex]. The curve passes through point [latex]\\left(0,0\\right)[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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49.\u00a0<\/strong>[latex]\\frac{dy}{d\\theta }={\\sin}^{4}\\left(\\pi \\theta \\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"229711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"229711\"][latex]\\frac{3\\theta }{8}-\\frac{1}{4\\pi }\\sin\\left(2\\pi \\theta \\right)+\\frac{1}{32\\pi }\\sin\\left(4\\pi \\theta \\right)+C=f\\left(x\\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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50.\u00a0<\/strong>Find the length of the curve [latex]y=\\text{ln}\\left(\\csc{x}\\right),\\frac{\\pi }{4}\\le x\\le \\frac{\\pi }{2}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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51.\u00a0<\/strong>Find the length of the curve [latex]y=\\text{ln}\\left(\\sin{x}\\right),\\frac{\\pi }{3}\\le x\\le \\frac{\\pi }{2}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"708871\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"708871\"][latex]\\text{ln}\\left(\\sqrt{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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52.\u00a0<\/strong>Find the volume generated by revolving the curve [latex]y=\\cos\\left(3x\\right)[\/latex] about the x<\/em>-axis, [latex]0\\le x\\le \\frac{\\pi }{36}[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use this information: The inner product of two functions f<\/em> and g<\/em> over [latex]\\left[a,b\\right][\/latex] is defined by [latex]f\\left(x\\right)\\cdot g\\left(x\\right)=\\langle f,g\\rangle ={\\displaystyle\\int }_{a}^{b}f\\cdot gdx[\/latex]. Two distinct functions f<\/em> and g<\/em> are said to be orthogonal if [latex]\\langle f,g\\rangle =0[\/latex].<\/p>\r\n\r\n

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53.\u00a0<\/strong>Show that [latex]\\left\\{\\sin\\left(2x\\right),\\cos\\left(3x\\right)\\right\\}[\/latex] are orthogonal over the interval [latex]\\left[\\text{-}\\pi ,\\pi \\right][\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"957095\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"957095\"][latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(2x\\right)\\cos\\left(3x\\right)dx=0[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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54.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(mx\\right)\\cos\\left(nx\\right)dx[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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55.\u00a0<\/strong>Integrate [latex]{y}^{\\prime }=\\sqrt{\\tan{x}}{\\sec}^{4}x[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"499339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"499339\"][latex]\\sqrt{\\tan\\left(x\\right)}x\\left(\\frac{8\\tan{x}}{21}+\\frac{2}{7}\\sec{x}^{2}\\tan{x}\\right)+C=f\\left(x\\right)[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.<\/p>\r\n\r\n

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56.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{456}x\\cos{x}dx[\/latex] or [latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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57.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{350}x{\\sec}^{2}xdx[\/latex] or [latex]\\displaystyle\\int {\\tan}^{350}x\\sec{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"532876\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"532876\"]The second integral is more difficult because the first integral is simply a u-substitution type.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Fill in the blank to make a true statement.<\/p>\n

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

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[latex]{\\cos}^{2}x[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]{\\sec}^{2}x - 1=\\_\\_\\_\\_\\_\\_\\_[\/latex]<\/div>\n<\/div>\n<\/div>\n

Use an identity to reduce the power of the trigonometric function to a trigonometric function raised to the first power.<\/p>\n

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3.\u00a0<\/strong>[latex]{\\sin}^{2}x=\\_\\_\\_\\_\\_\\_\\_[\/latex]<\/p>\n

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

Evaluate each of the following integrals by u<\/em>-substitution.<\/p>\n

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5.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}x\\cos{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{{\\sin}^{4}x}{4}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\cos{x}}\\sin{x}dx[\/latex]<\/div>\n
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7.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{5}\\left(2x\\right){\\sec}^{2}\\left(2x\\right)dx[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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[latex]\\frac{1}{12}{\\tan}^{6}\\left(2x\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{7}\\left(2x\\right)\\cos\\left(2x\\right)dx[\/latex]<\/div>\n
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9.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan\\left(\\frac{x}{2}\\right){\\sec}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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[latex]{\\sec}^{2}\\left(\\frac{x}{2}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
10.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{2}x{\\sec}^{2}xdx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note<\/em>: Some of the problems may be done using techniques of integration learned previously.)<\/p>\n

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11.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{3}xdx[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{-}\\cos{x}+\\frac{1}{3}\\cos^{2}x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{3}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]\\displaystyle\\int \\sin{x}\\cos{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]-\\frac{1}{2}{\\cos}^{2}x+C[\/latex] or [latex]\\frac{1}{2}{\\sin}^{2}x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
14.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{5}xdx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{5}x{\\cos}^{2}xdx[\/latex]<\/p>\n<\/div>\n

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[latex]-\\frac{1}{3}\\cos^{3}x+\\frac{2}{5}\\cos^{5}x-\\frac{1}{7}\\cos^{7}x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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16. <\/strong>[latex]\\displaystyle\\int {\\sin}^{3}x{\\cos}^{3}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\sin{x}}\\cos{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{2}{3}{\\left(\\sin{x}\\right)}^{\\frac{3}{2}}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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18.\u00a0<\/strong>[latex]\\displaystyle\\int \\sqrt{\\sin{x}}{\\cos}^{3}xdx[\/latex]<\/div>\n
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19.\u00a0<\/strong>[latex]\\displaystyle\\int \\sec{x}\\tan{x}dx[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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[latex]\\sec{x}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan\\left(5x\\right)dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{2}x\\sec{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{2}\\sec{x}\\tan{x}-\\frac{1}{2}\\text{ln}\\left(\\sec{x}+\\tan{x}\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]\\displaystyle\\int \\tan{x}{\\sec}^{3}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sec}^{4}xdx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{2\\tan{x}}{3}+\\frac{1}{3}\\sec{\\left(x\\right)}^{2}\\tan{x}[\/latex] [latex]=\\tan{x}+\\frac{{\\tan}^{3}x}{3}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]\\displaystyle\\int \\cot{x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>[latex]\\displaystyle\\int \\csc{x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\text{-}\\text{ln}|\\cot{x}+\\csc{x}|+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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26.\u00a0<\/strong>[latex]\\displaystyle\\int \\frac{{\\tan}^{3}x}{\\sqrt{\\sec{x}}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, find a general formula for the integrals.<\/p>\n

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27.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}ax\\cos{ax} dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{{\\sin}^{3}\\left(ax\\right)}{3a}+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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28. <\/strong>[latex]\\displaystyle\\int \\sin{ax} \\cos{ax} dx[\/latex].<\/div>\n<\/div>\n<\/div>\n

Use the double-angle formulas to evaluate the following integrals.<\/p>\n

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29.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}xdx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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30.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{4}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{2}3xdx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{x}{2}+\\frac{1}{12}\\sin\\left(6x\\right)+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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32.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}xdx[\/latex]<\/div>\n
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33.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}xdx+\\displaystyle\\int {\\cos}^{2}xdx[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

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[latex]x+C[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}\\left(2x\\right)dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

For the following exercises, evaluate the definite integrals. Express answers in exact form whenever possible.<\/p>\n

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35.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2\\pi }\\cos{x}\\sin2xdx[\/latex]<\/p>\n<\/div>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }\\sin3x\\sin5xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\pi }\\cos\\left(99x\\right)\\sin\\left(101x\\right)dx[\/latex]<\/p>\n<\/div>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n
38.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }{\\cos}^{2}\\left(3x\\right)dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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39.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sin{x}\\sin\\left(2x\\right)\\sin\\left(3x\\right)dx[\/latex]<\/p>\n<\/div>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
40.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4\\pi }\\cos\\left(\\frac{x}{2}\\right)\\sin\\left(\\frac{x}{2}\\right)dx[\/latex]<\/span><\/div>\n
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41.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{\\pi}{6}}^{\\frac{\\pi}{3}}\\frac{{\\cos}^{3}x}{\\sqrt{\\sin{x}}}dx[\/latex] (Round this answer to three decimal places.)<\/span><\/p>\n<\/div>\n<\/div>\n

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Approximately 0.239<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
42.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\frac{\\text{-}\\pi}{3}}^{\\frac{\\pi}{3}}\\sqrt{{\\sec}^{2}x - 1}dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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43.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}\\sqrt{1-\\cos\\left(2x\\right)}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\sqrt{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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44.\u00a0<\/strong>Find the area of the region bounded by the graphs of the equations [latex]y=\\sin{x},y={\\sin}^{3}x,x=0,\\text{and }x=\\frac{\\pi }{2}[\/latex].<\/div>\n
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45.\u00a0<\/strong>Find the area of the region bounded by the graphs of the equations [latex]y={\\cos}^{2}x,y={\\sin}^{2}x,x=-\\frac{\\pi }{4},\\text{and }x=\\frac{\\pi }{4}[\/latex].<\/span><\/p>\n<\/div>\n<\/div>\n

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1.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
46.\u00a0<\/strong>A particle moves in a straight line with the velocity function [latex]v\\left(t\\right)=\\sin\\left(\\omega t\\right){\\cos}^{2}\\left(\\omega t\\right)[\/latex]. Find its position function [latex]x=f\\left(t\\right)[\/latex] if [latex]f\\left(0\\right)=0[\/latex].<\/span><\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>Find the average value of the function [latex]f\\left(x\\right)={\\sin}^{2}x{\\cos}^{3}x[\/latex] over the interval [latex]\\left[\\text{-}\\pi ,\\pi \\right][\/latex].<\/p>\n<\/div>\n

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0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, solve the differential equations.<\/span><\/p>\n<\/div>\n<\/div>\n

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48.\u00a0<\/strong>[latex]\\frac{dy}{dx}={\\sin}^{2}x[\/latex]. The curve passes through point [latex]\\left(0,0\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>[latex]\\frac{dy}{d\\theta }={\\sin}^{4}\\left(\\pi \\theta \\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{3\\theta }{8}-\\frac{1}{4\\pi }\\sin\\left(2\\pi \\theta \\right)+\\frac{1}{32\\pi }\\sin\\left(4\\pi \\theta \\right)+C=f\\left(x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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50.\u00a0<\/strong>Find the length of the curve [latex]y=\\text{ln}\\left(\\csc{x}\\right),\\frac{\\pi }{4}\\le x\\le \\frac{\\pi }{2}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>Find the length of the curve [latex]y=\\text{ln}\\left(\\sin{x}\\right),\\frac{\\pi }{3}\\le x\\le \\frac{\\pi }{2}[\/latex].<\/p>\n<\/div>\n

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[latex]\\text{ln}\\left(\\sqrt{3}\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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52.\u00a0<\/strong>Find the volume generated by revolving the curve [latex]y=\\cos\\left(3x\\right)[\/latex] about the x<\/em>-axis, [latex]0\\le x\\le \\frac{\\pi }{36}[\/latex].<\/div>\n<\/div>\n<\/div>\n

For the following exercises, use this information: The inner product of two functions f<\/em> and g<\/em> over [latex]\\left[a,b\\right][\/latex] is defined by [latex]f\\left(x\\right)\\cdot g\\left(x\\right)=\\langle f,g\\rangle ={\\displaystyle\\int }_{a}^{b}f\\cdot gdx[\/latex]. Two distinct functions f<\/em> and g<\/em> are said to be orthogonal if [latex]\\langle f,g\\rangle =0[\/latex].<\/p>\n

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53.\u00a0<\/strong>Show that [latex]\\left\\{\\sin\\left(2x\\right),\\cos\\left(3x\\right)\\right\\}[\/latex] are orthogonal over the interval [latex]\\left[\\text{-}\\pi ,\\pi \\right][\/latex].<\/p>\n<\/div>\n

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[latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(2x\\right)\\cos\\left(3x\\right)dx=0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(mx\\right)\\cos\\left(nx\\right)dx[\/latex].<\/div>\n<\/div>\n<\/div>\n
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55.\u00a0<\/strong>Integrate [latex]{y}^{\\prime }=\\sqrt{\\tan{x}}{\\sec}^{4}x[\/latex].<\/p>\n<\/div>\n

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[latex]\\sqrt{\\tan\\left(x\\right)}x\\left(\\frac{8\\tan{x}}{21}+\\frac{2}{7}\\sec{x}^{2}\\tan{x}\\right)+C=f\\left(x\\right)[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning.<\/p>\n

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56.\u00a0<\/strong>[latex]\\displaystyle\\int {\\sin}^{456}x\\cos{x}dx[\/latex] or [latex]\\displaystyle\\int {\\sin}^{2}x{\\cos}^{2}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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57.\u00a0<\/strong>[latex]\\displaystyle\\int {\\tan}^{350}x{\\sec}^{2}xdx[\/latex] or [latex]\\displaystyle\\int {\\tan}^{350}x\\sec{x}dx[\/latex]<\/p>\n<\/div>\n

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The second integral is more difficult because the first integral is simply a u-substitution type.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t
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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>