{"id":178,"date":"2026-01-12T16:00:34","date_gmt":"2026-01-12T16:00:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=178"},"modified":"2026-01-12T16:00:34","modified_gmt":"2026-01-12T16:00:34","slug":"integration-of-exponential-logarithmic-and-hyperbolic-functions-get-stronger-answer-key-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/integration-of-exponential-logarithmic-and-hyperbolic-functions-get-stronger-answer-key-2\/","title":{"raw":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Get Stronger Answer Key","rendered":"Integration of Exponential, Logarithmic, and Hyperbolic Functions: Get Stronger Answer Key"},"content":{"raw":"

Integrals, Exponential Functions, and Logarithms<\/h2>\r\n
    \r\n \t
  1. [latex]\\frac{1}{x}[\/latex]<\/li>\r\n \t
  2. [latex]-\\frac{1}{x{(\\text{ln}x)}^{2}}[\/latex]<\/li>\r\n \t
  3. [latex]\\text{ln}(x+1)+C[\/latex]<\/li>\r\n \t
  4. [latex]\\text{ln}(x)+1[\/latex]<\/li>\r\n \t
  5. [latex] \\cot (x)[\/latex]<\/li>\r\n \t
  6. [latex]\\frac{7}{x}[\/latex]<\/li>\r\n \t
  7. [latex] \\csc (x) \\sec x[\/latex]<\/li>\r\n \t
  8. [latex]-2 \\tan x[\/latex]<\/li>\r\n \t
  9. [latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/li>\r\n \t
  10. [latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/li>\r\n \t
  11. [latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]<\/li>\r\n \t
  12. [latex]\\frac{1}{2}\\text{ln}(2)[\/latex]<\/li>\r\n \t
  13. [latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/li>\r\n \t
  14. [latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/li>\r\n \t
  15. [latex]{x}^{-2-(1\\text{\/}x)}(\\text{ln}x-1)[\/latex]<\/li>\r\n \t
  16. [latex]e{x}^{e-1}[\/latex]<\/li>\r\n \t
  17. [latex]1[\/latex]<\/li>\r\n \t
  18. [latex]-\\frac{1}{{x}^{2}}[\/latex]<\/li>\r\n \t
  19. [latex]\\pi -\\text{ln}(2)[\/latex]<\/li>\r\n \t
  20. [latex]\\frac{1}{x}[\/latex]<\/li>\r\n \t
  21. [latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/li>\r\n \t
  22. [latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/li>\r\n \t
  23. <\/li>\r\n \t
  24. <\/li>\r\n \t
  25. <\/li>\r\n \t
  26. <\/li>\r\n \t
  27. <\/li>\r\n<\/ol>\r\n

    Exponential Growth and Decay<\/h2>\r\n
      \r\n \t
    1. True<\/li>\r\n \t
    2. False; [latex]k=\\frac{\\text{ln}(2)}{t}[\/latex]<\/li>\r\n \t
    3. [latex]20[\/latex] hours<\/li>\r\n \t
    4. No. The relic is approximately [latex]871[\/latex] years old.<\/li>\r\n \t
    5. [latex]71.92[\/latex] years<\/li>\r\n \t
    6. [latex]5[\/latex] days [latex]6[\/latex] hours [latex]27[\/latex] minutes<\/li>\r\n \t
    7. [latex]12[\/latex]<\/li>\r\n \t
    8. [latex]8.618\\%[\/latex]<\/li>\r\n \t
    9. [latex]$6766.76[\/latex]<\/li>\r\n \t
    10. [latex]9[\/latex] hours [latex]13[\/latex] minutes<\/li>\r\n \t
    11. [latex]239,179[\/latex] years<\/li>\r\n \t
    12. <\/li>\r\n \t
    13. [latex]P\\prime (t)=43{e}^{0.01604t}.[\/latex] The population is always increasing.<\/li>\r\n \t
    14. <\/li>\r\n \t
    15. The population reaches [latex]10[\/latex] billion people in 2027.<\/li>\r\n \t
    16. <\/li>\r\n \t
    17. [latex]P\\prime (t)=2.259{e}^{0.06407t}.[\/latex] The population is always increasing.<\/li>\r\n \t
    18. <\/li>\r\n<\/ol>\r\n

      Calculus of the Hyperbolic Functions<\/h2>\r\n
        \r\n \t
      1. [latex]{e}^{x}\\text{ and }{e}^{\\text{\u2212}x}[\/latex]<\/li>\r\n \t
      2. Answers may vary<\/li>\r\n \t
      3. Answers may vary<\/li>\r\n \t
      4. Answers may vary<\/li>\r\n \t
      5. [latex]3\\text{sinh}(3x+1)[\/latex]<\/li>\r\n \t
      6. [latex]\\text{\u2212}\\text{tanh}(x)\\text{sech}(x)[\/latex]<\/li>\r\n \t
      7. [latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/li>\r\n \t
      8. [latex]\\frac{x{\\text{sech}}^{2}(\\sqrt{{x}^{2}+1})}{\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\r\n \t
      9. [latex]6{\\text{sinh}}^{5}(x)\\text{cosh}(x)[\/latex]<\/li>\r\n \t
      10. [latex]\\frac{1}{2}\\text{sinh}(2x+1)+C[\/latex]<\/li>\r\n \t
      11. [latex]\\frac{1}{2}{\\text{sinh}}^{2}({x}^{2})+C[\/latex]<\/li>\r\n \t
      12. [latex]\\frac{1}{3}{\\text{cosh}}^{3}(x)+C[\/latex]<\/li>\r\n \t
      13. [latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex]]<\/li>\r\n \t
      14. [latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/li>\r\n \t
      15. [latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/li>\r\n \t
      16. [latex]\\frac{\\text{sinh}(x)}{\\sqrt{{\\text{cosh}}^{2}(x)+1}}[\/latex]<\/li>\r\n \t
      17. [latex]\\text{\u2212} \\csc (x)[\/latex]<\/li>\r\n \t
      18. [latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex]<\/li>\r\n \t
      19. [latex]\\frac{1}{a}{\\text{tanh}}^{-1}(\\frac{x}{a})+C[\/latex]<\/li>\r\n \t
      20. [latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/li>\r\n \t
      21. [latex]{\\text{cosh}}^{-1}({e}^{x})+C[\/latex]<\/li>\r\n \t
      22. [latex]-0.521095[\/latex]<\/li>\r\n \t
      23. [latex]10[\/latex]<\/li>\r\n \t
      24. <\/li>\r\n \t
      25. <\/li>\r\n \t
      26. <\/li>\r\n<\/ol>","rendered":"

        Integrals, Exponential Functions, and Logarithms<\/h2>\n
          \n
        1. [latex]\\frac{1}{x}[\/latex]<\/li>\n
        2. [latex]-\\frac{1}{x{(\\text{ln}x)}^{2}}[\/latex]<\/li>\n
        3. [latex]\\text{ln}(x+1)+C[\/latex]<\/li>\n
        4. [latex]\\text{ln}(x)+1[\/latex]<\/li>\n
        5. [latex]\\cot (x)[\/latex]<\/li>\n
        6. [latex]\\frac{7}{x}[\/latex]<\/li>\n
        7. [latex]\\csc (x) \\sec x[\/latex]<\/li>\n
        8. [latex]-2 \\tan x[\/latex]<\/li>\n
        9. [latex]\\frac{1}{2}\\text{ln}(\\frac{5}{3})[\/latex]<\/li>\n
        10. [latex]2-\\frac{1}{2}\\text{ln}(5)[\/latex]<\/li>\n
        11. [latex]\\frac{1}{\\text{ln}(2)}-1[\/latex]<\/li>\n
        12. [latex]\\frac{1}{2}\\text{ln}(2)[\/latex]<\/li>\n
        13. [latex]\\frac{1}{3}{(\\text{ln}x)}^{3}[\/latex]<\/li>\n
        14. [latex]\\frac{2{x}^{3}}{\\sqrt{{x}^{2}+1}\\sqrt{{x}^{2}-1}}[\/latex]<\/li>\n
        15. [latex]{x}^{-2-(1\\text{\/}x)}(\\text{ln}x-1)[\/latex]<\/li>\n
        16. [latex]e{x}^{e-1}[\/latex]<\/li>\n
        17. [latex]1[\/latex]<\/li>\n
        18. [latex]-\\frac{1}{{x}^{2}}[\/latex]<\/li>\n
        19. [latex]\\pi -\\text{ln}(2)[\/latex]<\/li>\n
        20. [latex]\\frac{1}{x}[\/latex]<\/li>\n
        21. [latex]{e}^{5}-6{\\text{units}}^{2}[\/latex]<\/li>\n
        22. [latex]\\text{ln}(4)-1{\\text{units}}^{2}[\/latex]<\/li>\n
        23. <\/li>\n
        24. <\/li>\n
        25. <\/li>\n
        26. <\/li>\n
        27. <\/li>\n<\/ol>\n

          Exponential Growth and Decay<\/h2>\n
            \n
          1. True<\/li>\n
          2. False; [latex]k=\\frac{\\text{ln}(2)}{t}[\/latex]<\/li>\n
          3. [latex]20[\/latex] hours<\/li>\n
          4. No. The relic is approximately [latex]871[\/latex] years old.<\/li>\n
          5. [latex]71.92[\/latex] years<\/li>\n
          6. [latex]5[\/latex] days [latex]6[\/latex] hours [latex]27[\/latex] minutes<\/li>\n
          7. [latex]12[\/latex]<\/li>\n
          8. [latex]8.618\\%[\/latex]<\/li>\n
          9. [latex]$6766.76[\/latex]<\/li>\n
          10. [latex]9[\/latex] hours [latex]13[\/latex] minutes<\/li>\n
          11. [latex]239,179[\/latex] years<\/li>\n
          12. <\/li>\n
          13. [latex]P\\prime (t)=43{e}^{0.01604t}.[\/latex] The population is always increasing.<\/li>\n
          14. <\/li>\n
          15. The population reaches [latex]10[\/latex] billion people in 2027.<\/li>\n
          16. <\/li>\n
          17. [latex]P\\prime (t)=2.259{e}^{0.06407t}.[\/latex] The population is always increasing.<\/li>\n
          18. <\/li>\n<\/ol>\n

            Calculus of the Hyperbolic Functions<\/h2>\n
              \n
            1. [latex]{e}^{x}\\text{ and }{e}^{\\text{\u2212}x}[\/latex]<\/li>\n
            2. Answers may vary<\/li>\n
            3. Answers may vary<\/li>\n
            4. Answers may vary<\/li>\n
            5. [latex]3\\text{sinh}(3x+1)[\/latex]<\/li>\n
            6. [latex]\\text{\u2212}\\text{tanh}(x)\\text{sech}(x)[\/latex]<\/li>\n
            7. [latex]4\\text{cosh}(x)\\text{sinh}(x)[\/latex]<\/li>\n
            8. [latex]\\frac{x{\\text{sech}}^{2}(\\sqrt{{x}^{2}+1})}{\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\n
            9. [latex]6{\\text{sinh}}^{5}(x)\\text{cosh}(x)[\/latex]<\/li>\n
            10. [latex]\\frac{1}{2}\\text{sinh}(2x+1)+C[\/latex]<\/li>\n
            11. [latex]\\frac{1}{2}{\\text{sinh}}^{2}({x}^{2})+C[\/latex]<\/li>\n
            12. [latex]\\frac{1}{3}{\\text{cosh}}^{3}(x)+C[\/latex]<\/li>\n
            13. [latex]\\text{ln}(1+\\text{cosh}(x))+C[\/latex]]<\/li>\n
            14. [latex]\\text{cosh}(x)+\\text{sinh}(x)+C[\/latex]<\/li>\n
            15. [latex]\\frac{4}{1-16{x}^{2}}[\/latex]<\/li>\n
            16. [latex]\\frac{\\text{sinh}(x)}{\\sqrt{{\\text{cosh}}^{2}(x)+1}}[\/latex]<\/li>\n
            17. [latex]\\text{\u2212} \\csc (x)[\/latex]<\/li>\n
            18. [latex]-\\frac{1}{({x}^{2}-1){\\text{tanh}}^{-1}(x)}[\/latex]<\/li>\n
            19. [latex]\\frac{1}{a}{\\text{tanh}}^{-1}(\\frac{x}{a})+C[\/latex]<\/li>\n
            20. [latex]\\sqrt{{x}^{2}+1}+C[\/latex]<\/li>\n
            21. [latex]{\\text{cosh}}^{-1}({e}^{x})+C[\/latex]<\/li>\n
            22. [latex]-0.521095[\/latex]<\/li>\n
            23. [latex]10[\/latex]<\/li>\n
            24. <\/li>\n
            25. <\/li>\n
            26. <\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":109,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/178"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/178\/revisions"}],"predecessor-version":[{"id":190,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/178\/revisions\/190"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/109"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/178\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=178"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=178"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=178"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}