{"id":71,"date":"2021-03-25T02:20:51","date_gmt":"2021-03-25T02:20:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/improper-integrals-2\/"},"modified":"2021-11-17T02:39:13","modified_gmt":"2021-11-17T02:39:13","slug":"improper-integrals-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus2\/chapter\/improper-integrals-2\/","title":{"raw":"Problem Set: Improper Integrals","rendered":"Problem Set: Improper Integrals"},"content":{"raw":"

Evaluate the following integrals. If the integral is not convergent, answer \"divergent.\"<\/p>\r\n\r\n

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

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[reveal-answer q=\"110743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"110743\"]divergent[\/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]{\\displaystyle\\int }_{0}^{\\infty }\\frac{1}{4+{x}^{2}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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3.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{\\sqrt{4-{x}^{2}}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"970811\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"970811\"][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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4.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{x\\text{ln}x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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5.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"778558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"778558\"][latex]\\frac{2}{e}[\/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 }_{\\text{-}\\infty }^{\\infty }\\frac{x}{{x}^{2}+1}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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7.\u00a0<\/strong>Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{\\sqrt{{x}^{3}+1}}dx[\/latex] converges or diverges by comparing the function [latex]f\\left(x\\right)=\\frac{1}{\\sqrt{{x}^{3}+1}}[\/latex] with [latex]g\\left(x\\right)=\\frac{1}{\\sqrt{{x}^{3}}}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"432156\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"432156\"]Converges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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8.\u00a0<\/strong>Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{\\sqrt{x+1}}dx[\/latex] converges or diverges.<\/div>\r\n<\/div>\r\n<\/div>\r\n

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.<\/p>\r\n\r\n

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9.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}\\cos{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"676338\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"676338\"]Converges to [latex]\\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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10.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{\\text{ln}x}{x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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11.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"843539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"843539\"]\u22124[\/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 }_{0}^{1}\\text{ln}xdx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{1}{{x}^{2}+1}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"637570\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637570\"][latex]\\pi [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{5}\\frac{dx}{\\sqrt{x - 1}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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15.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-2}^{2}\\frac{dx}{{\\left(1+x\\right)}^{2}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"475778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"475778\"]diverges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }\\sin{x}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"146897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"146897\"]diverges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{{e}^{x}}{1+{e}^{2x}}dx[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{\\sqrt[3]{x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"110113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"110113\"]1.5[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\frac{dx}{{x}^{3}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{2}\\frac{dx}{{x}^{3}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"29167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"29167\"]diverges[\/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 }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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23.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{x - 1}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"833586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"833586\"]diverges[\/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 }_{1}^{\\infty }\\frac{5}{{x}^{3}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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25.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{3}^{5}\\frac{5}{{\\left(x - 4\\right)}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"536236\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"536236\"]diverges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.<\/p>\r\n\r\n

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

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[reveal-answer q=\"145139\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145139\"]Both integrals diverge.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

Evaluate the integrals. If the integral diverges, answer \"diverges.\"<\/p>\r\n\r\n

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28.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{{x}^{e}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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29.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{{x}^{\\pi }}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"390175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"390175\"]diverges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

30.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-x}}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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31.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{1-x}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"288777\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"288777\"]diverges[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

32. <\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{dx}{{x}^{2}+1}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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33.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"741497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"741497\"][latex]\\pi [\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{\\text{ln}x}{x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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35.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{e}\\text{ln}\\left(x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"164855\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"164855\"]0.0[\/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}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{x}{{\\left({x}^{2}+1\\right)}^{2}}dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"756617\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"756617\"]0.0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n

Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.<\/p>\r\n\r\n

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39.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{9}\\frac{dx}{\\sqrt{9-x}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"469949\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"469949\"]6.0[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

40.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-27}^{1}\\frac{dx}{{x}^{\\frac{2}{3}}}[\/latex]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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41.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{dx}{\\sqrt{9-{x}^{2}}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"789307\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"789307\"][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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42.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{6}^{24}\\frac{dt}{t\\sqrt{{t}^{2}-36}}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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43.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4}x\\text{ln}\\left(4x\\right)dx[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"713735\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"713735\"][latex]8\\text{ln}\\left(16\\right)-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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44. <\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{x}{\\sqrt{9-{x}^{2}}}dx[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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45.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{.5}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]. (Be careful!) (Express your answer using three decimal places.)<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"432852\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"432852\"][latex]1.047[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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46.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{1}^{4}\\frac{dx}{\\sqrt{{x}^{2}-1}}[\/latex]. (Express the answer in exact form.)<\/div>\r\n<\/div>\r\n<\/div>\r\n
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47.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{2}^{\\infty }\\frac{dx}{{\\left({x}^{2}-1\\right)}^{\\frac{3}{2}}}[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"797846\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797846\"][latex]-1+\\frac{2}{\\sqrt{3}}[\/latex][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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48.\u00a0<\/strong>Find the area of the region in the first quadrant between the curve [latex]y={e}^{-6x}[\/latex] and the x<\/em>-axis.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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49.\u00a0<\/strong>Find the area of the region bounded by the curve [latex]y=\\frac{7}{{x}^{2}}[\/latex], the x<\/em>-axis, and on the left by [latex]x=1[\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"569515\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"569515\"]7.0[\/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 area under the curve [latex]y=\\frac{1}{{\\left(x+1\\right)}^{\\frac{3}{2}}}[\/latex], bounded on the left by [latex]x=3[\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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51.\u00a0<\/strong>Find the area under [latex]y=\\frac{5}{1+{x}^{2}}[\/latex] in the first quadrant.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"297999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"297999\"][latex]\\frac{5\\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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52.\u00a0<\/strong>Find the volume of the solid generated by revolving about the x<\/em>-axis the region under the curve [latex]y=\\frac{3}{x}[\/latex] from [latex]x=1[\/latex] to [latex]x=\\infty [\/latex].<\/div>\r\n<\/div>\r\n<\/div>\r\n
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53.\u00a0<\/strong>Find the volume of the solid generated by revolving about the y<\/em>-axis the region under the curve [latex]y=6{e}^{-2x}[\/latex] in the first quadrant.<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"61192\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"61192\"][latex]3\\pi [\/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>Find the volume of the solid generated by revolving about the x<\/em>-axis the area under the curve [latex]y=3{e}^{\\text{-}x}[\/latex] in the first quadrant.<\/div>\r\n<\/div>\r\n<\/div>\r\n

The Laplace transform<\/span> of a continuous function over the interval [latex]\\left[0,\\infty \\right)[\/latex] is defined by [latex]F\\left(s\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}sx}f\\left(x\\right)dx[\/latex] (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F<\/em> is the set of all real numbers s<\/em> such that the improper integral converges. Find the Laplace transform F<\/em> of each of the following functions and give the domain of F<\/em>.<\/p>\r\n\r\n

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55.\u00a0<\/strong>[latex]f\\left(x\\right)=1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"499171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"499171\"][latex]\\frac{1}{s},s>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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56.\u00a0<\/strong>[latex]f\\left(x\\right)=x[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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57.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"622373\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622373\"][latex]\\frac{s}{{s}^{2}+4},s>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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58.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{ax}[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n
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59.\u00a0<\/strong>Use the formula for arc length to show that the circumference of the circle [latex]{x}^{2}+{y}^{2}=1[\/latex] is [latex]2\\pi [\/latex].<\/p>\r\n\r\n<\/div>\r\n

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[reveal-answer q=\"754361\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"754361\"]Answers will vary.[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

A non-negative function is a probability density function<\/span> if it satisfies the following definition: [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }f\\left(t\\right)dt=1[\/latex]. The probability that a random variable x<\/em> lies between a<\/em> and b<\/em> is given by [latex]P\\left(a\\le x\\le b\\right)={\\displaystyle\\int }_{a}^{b}f\\left(t\\right)dt[\/latex].<\/p>\r\n\r\n

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60.\u00a0<\/strong>Show that [latex]f\\left(x\\right)=\\{\\begin{array}{c}0\\text{ if }x<0\\\\ 7{e}^{-7x}\\text{ if }x\\ge 0\\end{array}[\/latex] is a probability density function.<\/div>\r\n<\/div>\r\n<\/div>\r\n
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61.\u00a0<\/strong>Find the probability that x<\/em> is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.<\/p>\r\n[reveal-answer q=\"361050\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"361050\"]0.8775[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

Evaluate the following integrals. If the integral is not convergent, answer “divergent.”<\/p>\n

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

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divergent<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }\\frac{1}{4+{x}^{2}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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3.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\frac{1}{\\sqrt{4-{x}^{2}}}dx[\/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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4.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{x\\text{ln}x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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5.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{2}{e}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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6.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{x}{{x}^{2}+1}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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7.\u00a0<\/strong>Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{\\sqrt{{x}^{3}+1}}dx[\/latex] converges or diverges by comparing the function [latex]f\\left(x\\right)=\\frac{1}{\\sqrt{{x}^{3}+1}}[\/latex] with [latex]g\\left(x\\right)=\\frac{1}{\\sqrt{{x}^{3}}}[\/latex].<\/p>\n<\/div>\n

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Converges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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8.\u00a0<\/strong>Without integrating, determine whether the integral [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{1}{\\sqrt{x+1}}dx[\/latex] converges or diverges.<\/div>\n<\/div>\n<\/div>\n

Determine whether the improper integrals converge or diverge. If possible, determine the value of the integrals that converge.<\/p>\n

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9.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}\\cos{x}dx[\/latex]<\/p>\n<\/div>\n

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Converges to [latex]\\frac{1}{2}[\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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10.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{\\text{ln}x}{x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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11.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex]<\/p>\n<\/div>\n

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\u22124<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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12.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\text{ln}xdx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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13.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{1}{{x}^{2}+1}dx[\/latex]<\/p>\n<\/div>\n

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[latex]\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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14.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{5}\\frac{dx}{\\sqrt{x - 1}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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15.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-2}^{2}\\frac{dx}{{\\left(1+x\\right)}^{2}}[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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16.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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17.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }\\sin{x}dx[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
18.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{{e}^{x}}{1+{e}^{2x}}dx[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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19.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{\\sqrt[3]{x}}[\/latex]<\/p>\n<\/div>\n

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1.5<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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20.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{2}\\frac{dx}{{x}^{3}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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21.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{2}\\frac{dx}{{x}^{3}}[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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22.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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23.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{1}{x - 1}dx[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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24.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{5}{{x}^{3}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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25.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{3}^{5}\\frac{5}{{\\left(x - 4\\right)}^{2}}dx[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Determine the convergence of each of the following integrals by comparison with the given integral. If the integral converges, find the number to which it converges.<\/p>\n

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26.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{{x}^{2}+4x}[\/latex]; compare with [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{{x}^{2}}[\/latex].<\/div>\n<\/div>\n<\/div>\n
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27.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{\\sqrt{x}+1}[\/latex]; compare with [latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{2\\sqrt{x}}[\/latex].<\/p>\n<\/div>\n

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Both integrals diverge.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

Evaluate the integrals. If the integral diverges, answer “diverges.”<\/p>\n

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28.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{1}^{\\infty }\\frac{dx}{{x}^{e}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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29.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{{x}^{\\pi }}[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
30.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{\\sqrt{1-x}}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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31.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{dx}{1-x}[\/latex]<\/p>\n<\/div>\n

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diverges<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
32. <\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{0}\\frac{dx}{{x}^{2}+1}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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33.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-1}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]<\/p>\n<\/div>\n

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[latex]\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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34.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}\\frac{\\text{ln}x}{x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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35.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{e}\\text{ln}\\left(x\\right)dx[\/latex]<\/p>\n<\/div>\n

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0.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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36.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }x{e}^{\\text{-}x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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37.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }\\frac{x}{{\\left({x}^{2}+1\\right)}^{2}}dx[\/latex]<\/p>\n<\/div>\n

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0.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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38.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}x}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n

Evaluate the improper integrals. Each of these integrals has an infinite discontinuity either at an endpoint or at an interior point of the interval.<\/p>\n

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39.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{9}\\frac{dx}{\\sqrt{9-x}}[\/latex]<\/p>\n<\/div>\n

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6.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
40.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{-27}^{1}\\frac{dx}{{x}^{\\frac{2}{3}}}[\/latex]<\/span><\/div>\n<\/div>\n<\/div>\n
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41.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{dx}{\\sqrt{9-{x}^{2}}}[\/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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42.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{6}^{24}\\frac{dt}{t\\sqrt{{t}^{2}-36}}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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43.\u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{4}x\\text{ln}\\left(4x\\right)dx[\/latex]<\/p>\n<\/div>\n

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[latex]8\\text{ln}\\left(16\\right)-4[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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44. <\/strong>[latex]{\\displaystyle\\int }_{0}^{3}\\frac{x}{\\sqrt{9-{x}^{2}}}dx[\/latex]<\/div>\n<\/div>\n<\/div>\n
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45.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{.5}^{1}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]. (Be careful!) (Express your answer using three decimal places.)<\/p>\n<\/div>\n

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[latex]1.047[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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46.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{1}^{4}\\frac{dx}{\\sqrt{{x}^{2}-1}}[\/latex]. (Express the answer in exact form.)<\/div>\n<\/div>\n<\/div>\n
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47.\u00a0<\/strong>Evaluate [latex]{\\displaystyle\\int }_{2}^{\\infty }\\frac{dx}{{\\left({x}^{2}-1\\right)}^{\\frac{3}{2}}}[\/latex].<\/p>\n<\/div>\n

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[latex]-1+\\frac{2}{\\sqrt{3}}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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48.\u00a0<\/strong>Find the area of the region in the first quadrant between the curve [latex]y={e}^{-6x}[\/latex] and the x<\/em>-axis.<\/div>\n<\/div>\n<\/div>\n
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49.\u00a0<\/strong>Find the area of the region bounded by the curve [latex]y=\\frac{7}{{x}^{2}}[\/latex], the x<\/em>-axis, and on the left by [latex]x=1[\/latex].<\/p>\n<\/div>\n

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7.0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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50.\u00a0<\/strong>Find the area under the curve [latex]y=\\frac{1}{{\\left(x+1\\right)}^{\\frac{3}{2}}}[\/latex], bounded on the left by [latex]x=3[\/latex].<\/div>\n<\/div>\n<\/div>\n
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51.\u00a0<\/strong>Find the area under [latex]y=\\frac{5}{1+{x}^{2}}[\/latex] in the first quadrant.<\/p>\n<\/div>\n

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[latex]\\frac{5\\pi }{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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52.\u00a0<\/strong>Find the volume of the solid generated by revolving about the x<\/em>-axis the region under the curve [latex]y=\\frac{3}{x}[\/latex] from [latex]x=1[\/latex] to [latex]x=\\infty[\/latex].<\/div>\n<\/div>\n<\/div>\n
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53.\u00a0<\/strong>Find the volume of the solid generated by revolving about the y<\/em>-axis the region under the curve [latex]y=6{e}^{-2x}[\/latex] in the first quadrant.<\/p>\n<\/div>\n

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[latex]3\\pi[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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54.\u00a0<\/strong>Find the volume of the solid generated by revolving about the x<\/em>-axis the area under the curve [latex]y=3{e}^{\\text{-}x}[\/latex] in the first quadrant.<\/div>\n<\/div>\n<\/div>\n

The Laplace transform<\/span> of a continuous function over the interval [latex]\\left[0,\\infty \\right)[\/latex] is defined by [latex]F\\left(s\\right)={\\displaystyle\\int }_{0}^{\\infty }{e}^{\\text{-}sx}f\\left(x\\right)dx[\/latex] (see the Student Project). This definition is used to solve some important initial-value problems in differential equations, as discussed later. The domain of F<\/em> is the set of all real numbers s<\/em> such that the improper integral converges. Find the Laplace transform F<\/em> of each of the following functions and give the domain of F<\/em>.<\/p>\n

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55.\u00a0<\/strong>[latex]f\\left(x\\right)=1[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{1}{s},s>0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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56.\u00a0<\/strong>[latex]f\\left(x\\right)=x[\/latex]<\/div>\n<\/div>\n<\/div>\n
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57.\u00a0<\/strong>[latex]f\\left(x\\right)=\\cos\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n

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[latex]\\frac{s}{{s}^{2}+4},s>0[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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58.\u00a0<\/strong>[latex]f\\left(x\\right)={e}^{ax}[\/latex]<\/div>\n<\/div>\n<\/div>\n
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59.\u00a0<\/strong>Use the formula for arc length to show that the circumference of the circle [latex]{x}^{2}+{y}^{2}=1[\/latex] is [latex]2\\pi[\/latex].<\/p>\n<\/div>\n

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Answers will vary.<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

A non-negative function is a probability density function<\/span> if it satisfies the following definition: [latex]{\\displaystyle\\int }_{\\text{-}\\infty }^{\\infty }f\\left(t\\right)dt=1[\/latex]. The probability that a random variable x<\/em> lies between a<\/em> and b<\/em> is given by [latex]P\\left(a\\le x\\le b\\right)={\\displaystyle\\int }_{a}^{b}f\\left(t\\right)dt[\/latex].<\/p>\n

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60.\u00a0<\/strong>Show that [latex]f\\left(x\\right)=\\{\\begin{array}{c}0\\text{ if }x<0\\\\ 7{e}^{-7x}\\text{ if }x\\ge 0\\end{array}[\/latex] is a probability density function.<\/div>\n<\/div>\n<\/div>\n
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61.\u00a0<\/strong>Find the probability that x<\/em> is between 0 and 0.3. (Use the function defined in the preceding problem.) Use four-place decimal accuracy.<\/p>\n

Show Solution<\/span><\/p>\n
0.8775<\/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>