{"id":179,"date":"2026-01-12T16:00:39","date_gmt":"2026-01-12T16:00:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=179"},"modified":"2026-01-12T16:00:39","modified_gmt":"2026-01-12T16:00:39","slug":"advanced-integration-techniques-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/advanced-integration-techniques-get-stronger-answer-key\/","title":{"raw":"Advanced Integration Techniques: Get Stronger Answer Key","rendered":"Advanced Integration Techniques: Get Stronger Answer Key"},"content":{"raw":"

Integration by Parts<\/h2>\r\n
    \r\n \t
  1. [latex]u={x}^{3}[\/latex]<\/li>\r\n \t
  2. [latex]u={y}^{3}[\/latex]<\/li>\r\n \t
  3. [latex]u=\\sin\\left(2x\\right)[\/latex]<\/li>\r\n \t
  4. [latex]\\text{-}x+x\\text{ln}x+C[\/latex]<\/li>\r\n \t
  5. [latex]x{\\tan}^{-1}x-\\frac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex]<\/li>\r\n \t
  6. [latex]-\\frac{1}{2}x\\cos\\left(2x\\right)+\\frac{1}{4}\\sin\\left(2x\\right)+C[\/latex]<\/li>\r\n \t
  7. [latex]{e}^{\\text{-}x}\\left(-1-x\\right)+C[\/latex]<\/li>\r\n \t
  8. [latex]2x\\cos{x}+\\left(-2+{x}^{2}\\right)\\sin{x}+C[\/latex]<\/li>\r\n \t
  9. [latex]\\frac{1}{2}\\left(1+2x\\right)\\left(-1+\\text{ln}\\left(1+2x\\right)\\right)+C[\/latex]<\/li>\r\n \t
  10. [latex]\\frac{1}{2}{e}^{x}\\left(\\text{-}\\cos{x}+\\sin{x}\\right)+C[\/latex]<\/li>\r\n \t
  11. [latex]-\\frac{{e}^{\\text{-}{x}^{2}}}{2}+C[\/latex]<\/li>\r\n \t
  12. [latex]-\\frac{1}{2}x\\cos\\left[\\text{ln}\\left(2x\\right)\\right]+\\frac{1}{2}x\\sin\\left[\\text{ln}\\left(2x\\right)\\right]+C[\/latex]<\/li>\r\n \t
  13. [latex]2x - 2x\\text{ln}x+x{\\left(\\text{ln}x\\right)}^{2}+C[\/latex]<\/li>\r\n \t
  14. [latex]\\left(\\text{-}\\frac{{x}^{3}}{9}+\\frac{1}{3}{x}^{3}\\text{ln}x\\right)+C[\/latex]<\/li>\r\n \t
  15. [latex]-\\frac{1}{2}\\sqrt{1 - 4{x}^{2}}+x{\\cos}^{-1}\\left(2x\\right)+C[\/latex]<\/li>\r\n \t
  16. [latex]\\text{-}\\left(-2+{x}^{2}\\right)\\cos{x}+2x\\sin{x}+C[\/latex]<\/li>\r\n \t
  17. [latex]\\text{-}x\\left(-6+{x}^{2}\\right)\\cos{x}+3\\left(-2+{x}^{2}\\right)\\sin{x}+C[\/latex]<\/li>\r\n \t
  18. [latex]\\frac{1}{2}x\\left(\\text{-}\\sqrt{1-\\frac{1}{{x}^{2}}}+x\\cdot {\\sec}^{-1}x\\right)+C[\/latex]<\/li>\r\n \t
  19. [latex]\\text{-}\\text{cosh}x+x\\text{sinh}x+C[\/latex]<\/li>\r\n \t
  20. [latex]\\frac{1}{4}-\\frac{3}{4{\\text{e}}^{2}}[\/latex]<\/li>\r\n \t
  21. [latex]2[\/latex]<\/li>\r\n \t
  22. [latex]2\\pi [\/latex]<\/li>\r\n \t
  23. [latex]-2+\\pi [\/latex]<\/li>\r\n \t
  24. [latex]\\text{-}\\sin\\left(x\\right)+\\text{ln}\\left[\\sin\\left(x\\right)\\right]\\sin{x}+C[\/latex]<\/li>\r\n \t
  25. Answers vary\r\n
      \r\n \t
    1. [latex]\\frac{2}{5}\\left(1+x\\right){\\left(-3+2x\\right)}^{\\frac{3}{2}}+C[\/latex]<\/li>\r\n \t
    2. [latex]\\frac{2}{5}\\left(1+x\\right){\\left(-3+2x\\right)}^{\\frac{3}{2}}+C[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
    3. Do not use integration by parts. Choose [latex]u[\/latex] to be [latex]\\text{ln}x[\/latex], and the integral is of the form [latex]\\displaystyle\\int {u}^{2}du[\/latex].<\/li>\r\n \t
    4. Do not use integration by parts. Let [latex]u={x}^{2}-3[\/latex], and the integral can be put into the form [latex]\\displaystyle\\int {e}^{u}du[\/latex].<\/li>\r\n \t
    5. Do not use integration by parts. Choose [latex]u[\/latex] to be [latex]u=3{x}^{3}+2[\/latex] and the integral can be put into the form [latex]\\displaystyle\\int \\sin\\left(u\\right)du[\/latex].<\/li>\r\n \t
    6. \r\n

      The area under graph is [latex]0.39535[\/latex].\r\n<\/span><\/p>\r\n\r\n\"This<\/span><\/li>\r\n \t

    7. [latex]2\\pi e[\/latex]<\/li>\r\n \t
    8. [latex]2.05[\/latex]<\/li>\r\n \t
    9. [latex]12\\pi [\/latex]<\/li>\r\n \t
    10. [latex]8{\\pi }^{2}[\/latex]<\/li>\r\n<\/ol>\r\n

      Trigonometric Integrals<\/h2>\r\n
        \r\n \t
      1. \r\n
          \r\n \t
        1. [latex]\\dfrac{{\\sin}^{4}x}{4}+C[\/latex]<\/li>\r\n \t
        2. [latex]\\dfrac{1}{12}{\\tan}^{6}\\left(2x\\right)+C[\/latex]<\/li>\r\n \t
        3. [latex]{\\sec}^{2}\\left(\\dfrac{x}{2}\\right)+C[\/latex]<\/li>\r\n \t
        4. [latex]\\text{-}\\cos{x}+\\dfrac{1}{3}\\cos^{2}x+C[\/latex]<\/li>\r\n \t
        5. [latex]-\\dfrac{1}{2}{\\cos}^{2}x+C[\/latex] or [latex]\\dfrac{1}{2}{\\sin}^{2}x+C[\/latex]<\/li>\r\n \t
        6. [latex]-\\dfrac{1}{3}\\cos^{3}x+\\dfrac{2}{5}\\cos^{5}x-\\dfrac{1}{7}\\cos^{7}x+C[\/latex]<\/li>\r\n \t
        7. [latex]\\dfrac{2}{3}{\\left(\\sin{x}\\right)}^{\\dfrac{3}{2}}+C[\/latex]<\/li>\r\n \t
        8. [latex]\\sec{x}+C[\/latex]<\/li>\r\n \t
        9. [latex]\\dfrac{1}{2}\\sec{x}\\tan{x}-\\dfrac{1}{2}\\text{ln}\\left(\\sec{x}+\\tan{x}\\right)+C[\/latex]<\/li>\r\n \t
        10. [latex]\\dfrac{2\\tan{x}}{3}+\\dfrac{1}{3}\\sec{\\left(x\\right)}^{2}\\tan{x}[\/latex] [latex]=\\tan{x}+\\dfrac{{\\tan}^{3}x}{3}+C[\/latex]<\/li>\r\n \t
        11. [latex]\\text{-}\\text{ln}|\\cot{x}+\\csc{x}|+C[\/latex]<\/li>\r\n \t
        12. [latex]\\dfrac{{\\sin}^{3}\\left(ax\\right)}{3a}+C[\/latex]<\/li>\r\n \t
        13. [latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t
        14. [latex]\\dfrac{x}{2}+\\dfrac{1}{12}\\sin\\left(6x\\right)+C[\/latex]<\/li>\r\n \t
        15. [latex]x+C[\/latex]<\/li>\r\n \t
        16. [latex]0[\/latex]<\/li>\r\n \t
        17. [latex]0[\/latex]<\/li>\r\n \t
        18. [latex]0[\/latex]<\/li>\r\n \t
        19. [latex]\\text{Approximately 0.239}[\/latex]<\/li>\r\n \t
        20. [latex]\\sqrt{2}[\/latex]<\/li>\r\n \t
        21. [latex]1.0[\/latex]<\/li>\r\n \t
        22. [latex]0[\/latex]<\/li>\r\n \t
        23. [latex]\\dfrac{3\\theta }{8}-\\dfrac{1}{4\\pi }\\sin\\left(2\\pi \\theta \\right)+\\dfrac{1}{32\\pi }\\sin\\left(4\\pi \\theta \\right)+C=f\\left(x\\right)[\/latex]<\/li>\r\n \t
        24. [latex]\\text{ln}\\left(\\sqrt{3}\\right)[\/latex]<\/li>\r\n \t
        25. [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(2x\\right)\\cos\\left(3x\\right)dx=0[\/latex]<\/li>\r\n \t
        26. [latex]\\sqrt{\\tan\\left(x\\right)}x\\left(\\dfrac{8\\tan{x}}{21}+\\dfrac{2}{7}\\sec{x}^{2}\\tan{x}\\right)+C=f\\left(x\\right)[\/latex]<\/li>\r\n \t
        27. The second integral is more difficult because the first integral is simply a u-substitution type.<\/li>\r\n \t
        28. [latex]9{\\tan}^{2}\\theta [\/latex]<\/li>\r\n \t
        29. [latex]{a}^{2}{\\text{cosh}}^{2}\\theta [\/latex]<\/li>\r\n \t
        30. [latex]4{\\left(x-\\dfrac{1}{2}\\right)}^{2}[\/latex]<\/li>\r\n \t
        31. [latex]\\text{-}{\\left(x+1\\right)}^{2}+5[\/latex]<\/li>\r\n \t
        32. [latex]\\text{ln}|x+\\sqrt{\\text{-}{a}^{2}+{x}^{2}}|+C[\/latex]<\/li>\r\n \t
        33. [latex]\\dfrac{1}{3}\\text{ln}|\\sqrt{9{x}^{2}+1}+3x|+C[\/latex]<\/li>\r\n \t
        34. [latex]-\\dfrac{\\sqrt{1-{x}^{2}}}{x}+C[\/latex]<\/li>\r\n \t
        35. [latex]9\\left[\\dfrac{x\\sqrt{{x}^{2}+9}}{18}+\\dfrac{1}{2}ln|\\dfrac{\\sqrt{{x}^{2}+9}}{3}+\\dfrac{x}{3}|\\right]+C[\/latex]<\/li>\r\n \t
        36. [latex]-\\dfrac{1}{3}\\sqrt{9-{\\theta }^{2}}\\left(18+{\\theta }^{2}\\right)+C[\/latex]<\/li>\r\n \t
        37. [latex]-\\dfrac{\\sqrt{1+{x}^{2}}}{x}+C[\/latex]<\/li>\r\n \t
        38. [latex]\\dfrac{1}{8}\\left(x\\left(5 - 2{x}^{2}\\right)\\sqrt{1-{x}^{2}}+3\\text{arcsin}x\\right)+C[\/latex]<\/li>\r\n \t
        39. [latex]\\text{ln}x-\\text{ln}|1+\\sqrt{1-{x}^{2}}|+C[\/latex]<\/li>\r\n \t
        40. [latex]-\\dfrac{\\sqrt{-1+{x}^{2}}}{x}+\\text{ln}|x+\\sqrt{-1+{x}^{2}}|+C[\/latex]<\/li>\r\n \t
        41. [latex]-\\dfrac{\\sqrt{1+{x}^{2}}}{x}+\\text{arcsinh}x+C[\/latex]<\/li>\r\n \t
        42. [latex]-\\dfrac{1}{1+x}+C[\/latex]<\/li>\r\n \t
        43. [latex]\\dfrac{2\\sqrt{-10+x}\\sqrt{x}\\text{ln}|\\sqrt{-10+x}+\\sqrt{x}|}{\\sqrt{\\left(10-x\\right)x}}+C[\/latex]<\/li>\r\n \t
        44. [latex]\\dfrac{9\\pi }{2}[\/latex]; area of a semicircle with radius 3<\/li>\r\n \t
        45. [latex]\\text{arcsin}\\left(x\\right)+C[\/latex] is the common answer.<\/li>\r\n \t
        46. [latex]\\dfrac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex] is the result using either method.<\/li>\r\n \t
        47. Use trigonometric substitution. Let [latex]x=\\sec\\left(\\theta \\right)[\/latex].<\/li>\r\n \t
        48. [latex]4.367[\/latex]<\/li>\r\n \t
        49. [latex]\\dfrac{{\\pi }^{2}}{8}+\\dfrac{\\pi }{4}[\/latex]<\/li>\r\n \t
        50. [latex]y=\\dfrac{1}{16}\\text{ln}|\\dfrac{x+8}{x - 8}|+3[\/latex]<\/li>\r\n \t
        51. [latex]24.6[\/latex] m\u00b3<\/li>\r\n \t
        52. [latex]\\dfrac{2\\pi }{3}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

          Partial Fractions<\/h2>\r\n
            \r\n \t
          1. [latex]-\\dfrac{2}{x+1}+\\dfrac{5}{2\\left(x+2\\right)}+\\dfrac{1}{2x}[\/latex]<\/li>\r\n \t
          2. [latex]\\dfrac{1}{{x}^{2}}+\\dfrac{3}{x}[\/latex]<\/li>\r\n \t
          3. [latex]2{x}^{2}+4x+8+\\dfrac{16}{x - 2}[\/latex]<\/li>\r\n \t
          4. [latex]-\\dfrac{1}{{x}^{2}}-\\dfrac{1}{x}+\\dfrac{1}{x - 1}[\/latex]<\/li>\r\n \t
          5. [latex]-\\dfrac{1}{2\\left(x - 2\\right)}+\\dfrac{1}{2\\left(x - 1\\right)}-\\dfrac{1}{6x}+\\dfrac{1}{6\\left(x - 3\\right)}[\/latex]<\/li>\r\n \t
          6. [latex]\\dfrac{1}{x - 1}+\\dfrac{2x+1}{{x}^{2}+x+1}[\/latex]<\/li>\r\n \t
          7. [latex]\\dfrac{2}{x+1}+\\dfrac{x}{{x}^{2}+4}-\\dfrac{1}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/li>\r\n \t
          8. [latex]\\text{-}\\text{ln}|2-x|+2\\text{ln}|4+x|+C[\/latex]<\/li>\r\n \t
          9. [latex]\\dfrac{1}{2}\\text{ln}|4-{x}^{2}|+C[\/latex]<\/li>\r\n \t
          10. [latex]2\\left(x+\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{1+x}{3}\\right)\\right)+C[\/latex]<\/li>\r\n \t
          11. [latex]2\\text{ln}|x|-3\\text{ln}|1+x|+C[\/latex]<\/li>\r\n \t
          12. [latex]\\dfrac{1}{16}\\left(\\text{-}\\dfrac{4}{-2+x}-\\text{ln}|-2+x|+\\text{ln}|2+x|\\right)+C[\/latex]<\/li>\r\n \t
          13. [latex]\\dfrac{1}{30}\\left(-2\\sqrt{5}\\text{arctan}\\left[\\dfrac{1+x}{\\sqrt{5}}\\right]+2\\text{ln}|-4+x|-\\text{ln}|6+2x+{x}^{2}|\\right)+C[\/latex]<\/li>\r\n \t
          14. [latex]-\\dfrac{3}{x}+4\\text{ln}|x+2|+x+C[\/latex]<\/li>\r\n \t
          15. [latex]\\text{-}\\text{ln}|3-x|+\\dfrac{1}{2}\\text{ln}|{x}^{2}+4|+C[\/latex]<\/li>\r\n \t
          16. [latex]\\text{ln}|x - 2|-\\dfrac{1}{2}\\text{ln}|{x}^{2}+2x+2|+C[\/latex]<\/li>\r\n \t
          17. [latex]\\text{-}x+\\text{ln}|1-{e}^{x}|+C[\/latex]<\/li>\r\n \t
          18. [latex]\\dfrac{1}{5}\\text{ln}|\\dfrac{\\cos{x}+3}{\\cos{x} - 2}|+C[\/latex]<\/li>\r\n \t
          19. [latex]\\dfrac{1}{2 - 2{e}^{2t}}+C[\/latex]<\/li>\r\n \t
          20. [latex]2\\sqrt{1+x}-2\\text{ln}|1+\\sqrt{1+x}|+C[\/latex]<\/li>\r\n \t
          21. [latex]\\text{ln}|\\dfrac{\\sin{x}}{1-\\sin{x}}|+C[\/latex]<\/li>\r\n \t
          22. [latex]\\dfrac{\\sqrt{3}}{4}[\/latex]<\/li>\r\n \t
          23. [latex]x-\\text{ln}\\left(1+{e}^{x}\\right)+C[\/latex]<\/li>\r\n \t
          24. [latex]6{x}^{\\dfrac{1}{6}}-3{x}^{\\dfrac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left(1+{x}^{\\dfrac{1}{6}}\\right)+C[\/latex]<\/li>\r\n \t
          25. [latex]\\dfrac{4}{3}\\pi \\text{arctanh}\\left[\\dfrac{1}{3}\\right]=\\dfrac{1}{3}\\pi \\text{ln}4[\/latex]<\/li>\r\n \t
          26. [latex]x=\\text{-}\\text{ln}|t - 3|+\\text{ln}|t - 4|+\\text{ln}2[\/latex]<\/li>\r\n \t
          27. [latex]x=\\text{ln}|t - 1|-\\sqrt{2}\\text{arctan}\\left(\\sqrt{2}t\\right)-\\dfrac{1}{2}\\text{ln}\\left({t}^{2}+\\dfrac{1}{2}\\right)+\\sqrt{2}\\text{arctan}\\left(2\\sqrt{2}\\right)+\\dfrac{1}{2}\\text{ln}4.5[\/latex]<\/li>\r\n \t
          28. [latex]\\dfrac{2}{5}\\pi \\text{ln}\\dfrac{28}{13}[\/latex]<\/li>\r\n \t
          29. [latex]\\dfrac{\\text{arctan}\\left[\\dfrac{-1+2x}{\\sqrt{3}}\\right]}{\\sqrt{3}}+\\dfrac{1}{3}\\text{ln}|1+x|-\\dfrac{1}{6}\\text{ln}|1-x+{x}^{2}|+C[\/latex]<\/li>\r\n \t
          30. [latex]2.0[\/latex] in.\u00b2<\/li>\r\n \t
          31. [latex]3{\\left(-8+x\\right)}^{\\dfrac{1}{3}}[\/latex] [latex]-2\\sqrt{3}\\text{arctan}\\left[\\dfrac{-1+{\\left(-8+x\\right)}^{\\dfrac{1}{3}}}{\\sqrt{3}}\\right][\/latex] [latex]-2\\text{ln}\\left[2+{\\left(-8+x\\right)}^{\\dfrac{1}{3}}\\right][\/latex] [latex]+\\text{ln}\\left[4 - 2{\\left(-8+x\\right)}^{\\dfrac{1}{3}}+{\\left(-8+x\\right)}^{\\dfrac{2}{3}}\\right]+C[\/latex]<\/li>\r\n<\/ol>\r\n

            Other Strategies for Integration<\/h2>\r\n
              \r\n \t
            1. [latex]\\dfrac{1}{2}\\text{ln}|{x}^{2}+2x+2|+2\\text{arctan}\\left(x+1\\right)+C[\/latex]<\/li>\r\n \t
            2. [latex]{\\text{cosh}}^{-1}\\left(\\dfrac{x+3}{3}\\right)+C[\/latex]<\/li>\r\n \t
            3. [latex]\\dfrac{{2}^{{x}^{2}-1}}{\\text{ln}2}+C[\/latex]<\/li>\r\n \t
            4. [latex]\\text{arcsin}\\left(\\dfrac{y}{2}\\right)+C[\/latex]<\/li>\r\n \t
            5. [latex]-\\dfrac{1}{2}\\csc\\left(2w\\right)+C[\/latex]<\/li>\r\n \t
            6. [latex]9 - 6\\sqrt{2}[\/latex]<\/li>\r\n \t
            7. [latex]2-\\dfrac{\\pi }{2}[\/latex]<\/li>\r\n \t
            8. [latex]\\dfrac{1}{12}{\\tan}^{4}\\left(3x\\right)-\\dfrac{1}{6}{\\tan}^{2}\\left(3x\\right)+\\dfrac{1}{3}\\text{ln}|\\sec\\left(3x\\right)|+C[\/latex]<\/li>\r\n \t
            9. [latex]2\\cot\\left(\\dfrac{w}{2}\\right)-2\\csc\\left(\\dfrac{w}{2}\\right)+w+C[\/latex]<\/li>\r\n \t
            10. [latex]\\dfrac{1}{5}\\text{ln}|\\dfrac{2\\left(5+4\\sin{t} - 3\\cos{t}\\right)}{4\\cos{t}+3\\sin{t}}|[\/latex]<\/li>\r\n \t
            11. [latex]6{x}^{\\dfrac{1}{6}}-3{x}^{\\dfrac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left[1+{x}^{\\dfrac{1}{6}}\\right]+C[\/latex]<\/li>\r\n \t
            12. [latex]\\text{-}{x}^{3}\\cos{x}+3{x}^{2}\\sin{x}+6x\\cos{x} - 6\\sin{x}+C[\/latex]<\/li>\r\n \t
            13. [latex]\\dfrac{1}{2}\\left({x}^{2}+\\text{ln}|1+{e}^{\\text{-}{x}^{2}}|\\right)+C[\/latex]<\/li>\r\n \t
            14. [latex]2\\text{arctan}\\left(\\sqrt{x - 1}\\right)+C[\/latex]<\/li>\r\n \t
            15. [latex]0.5=\\dfrac{1}{2}[\/latex]<\/li>\r\n \t
            16. [latex]8.0[\/latex]<\/li>\r\n \t
            17. [latex]\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{1}{3}\\left(x+2\\right)\\right)+C[\/latex]<\/li>\r\n \t
            18. [latex]\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{x+1}{3}\\right)+C[\/latex]<\/li>\r\n \t
            19. [latex]\\text{ln}\\left({e}^{x}+\\sqrt{4+{e}^{2x}}\\right)+C[\/latex]<\/li>\r\n \t
            20. [latex]\\text{ln}x-\\dfrac{1}{6}\\text{ln}\\left({x}^{6}+1\\right)-\\dfrac{\\text{arctan}\\left({x}^{3}\\right)}{3{x}^{3}}+C[\/latex]<\/li>\r\n \t
            21. [latex]\\text{ln}|x+\\sqrt{16+{x}^{2}}|+C[\/latex]<\/li>\r\n \t
            22. [latex]-\\dfrac{1}{4}\\cot\\left(2x\\right)+C[\/latex]<\/li>\r\n \t
            23. [latex]\\dfrac{1}{2}\\text{arctan}10[\/latex]<\/li>\r\n \t
            24. [latex]1276.14[\/latex]<\/li>\r\n \t
            25. [latex]7.21[\/latex]<\/li>\r\n \t
            26. [latex]\\sqrt{5}-\\sqrt{2}+\\text{ln}|\\dfrac{2+2\\sqrt{2}}{1+\\sqrt{5}}|[\/latex]<\/li>\r\n \t
            27. [latex]\\dfrac{1}{3}\\text{arctan}\\left(3\\right)\\approx 0.416[\/latex]<\/li>\r\n<\/ol>","rendered":"

              Integration by Parts<\/h2>\n
                \n
              1. [latex]u={x}^{3}[\/latex]<\/li>\n
              2. [latex]u={y}^{3}[\/latex]<\/li>\n
              3. [latex]u=\\sin\\left(2x\\right)[\/latex]<\/li>\n
              4. [latex]\\text{-}x+x\\text{ln}x+C[\/latex]<\/li>\n
              5. [latex]x{\\tan}^{-1}x-\\frac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex]<\/li>\n
              6. [latex]-\\frac{1}{2}x\\cos\\left(2x\\right)+\\frac{1}{4}\\sin\\left(2x\\right)+C[\/latex]<\/li>\n
              7. [latex]{e}^{\\text{-}x}\\left(-1-x\\right)+C[\/latex]<\/li>\n
              8. [latex]2x\\cos{x}+\\left(-2+{x}^{2}\\right)\\sin{x}+C[\/latex]<\/li>\n
              9. [latex]\\frac{1}{2}\\left(1+2x\\right)\\left(-1+\\text{ln}\\left(1+2x\\right)\\right)+C[\/latex]<\/li>\n
              10. [latex]\\frac{1}{2}{e}^{x}\\left(\\text{-}\\cos{x}+\\sin{x}\\right)+C[\/latex]<\/li>\n
              11. [latex]-\\frac{{e}^{\\text{-}{x}^{2}}}{2}+C[\/latex]<\/li>\n
              12. [latex]-\\frac{1}{2}x\\cos\\left[\\text{ln}\\left(2x\\right)\\right]+\\frac{1}{2}x\\sin\\left[\\text{ln}\\left(2x\\right)\\right]+C[\/latex]<\/li>\n
              13. [latex]2x - 2x\\text{ln}x+x{\\left(\\text{ln}x\\right)}^{2}+C[\/latex]<\/li>\n
              14. [latex]\\left(\\text{-}\\frac{{x}^{3}}{9}+\\frac{1}{3}{x}^{3}\\text{ln}x\\right)+C[\/latex]<\/li>\n
              15. [latex]-\\frac{1}{2}\\sqrt{1 - 4{x}^{2}}+x{\\cos}^{-1}\\left(2x\\right)+C[\/latex]<\/li>\n
              16. [latex]\\text{-}\\left(-2+{x}^{2}\\right)\\cos{x}+2x\\sin{x}+C[\/latex]<\/li>\n
              17. [latex]\\text{-}x\\left(-6+{x}^{2}\\right)\\cos{x}+3\\left(-2+{x}^{2}\\right)\\sin{x}+C[\/latex]<\/li>\n
              18. [latex]\\frac{1}{2}x\\left(\\text{-}\\sqrt{1-\\frac{1}{{x}^{2}}}+x\\cdot {\\sec}^{-1}x\\right)+C[\/latex]<\/li>\n
              19. [latex]\\text{-}\\text{cosh}x+x\\text{sinh}x+C[\/latex]<\/li>\n
              20. [latex]\\frac{1}{4}-\\frac{3}{4{\\text{e}}^{2}}[\/latex]<\/li>\n
              21. [latex]2[\/latex]<\/li>\n
              22. [latex]2\\pi[\/latex]<\/li>\n
              23. [latex]-2+\\pi[\/latex]<\/li>\n
              24. [latex]\\text{-}\\sin\\left(x\\right)+\\text{ln}\\left[\\sin\\left(x\\right)\\right]\\sin{x}+C[\/latex]<\/li>\n
              25. Answers vary\n
                  \n
                1. [latex]\\frac{2}{5}\\left(1+x\\right){\\left(-3+2x\\right)}^{\\frac{3}{2}}+C[\/latex]<\/li>\n
                2. [latex]\\frac{2}{5}\\left(1+x\\right){\\left(-3+2x\\right)}^{\\frac{3}{2}}+C[\/latex]<\/li>\n<\/ol>\n<\/li>\n
                3. Do not use integration by parts. Choose [latex]u[\/latex] to be [latex]\\text{ln}x[\/latex], and the integral is of the form [latex]\\displaystyle\\int {u}^{2}du[\/latex].<\/li>\n
                4. Do not use integration by parts. Let [latex]u={x}^{2}-3[\/latex], and the integral can be put into the form [latex]\\displaystyle\\int {e}^{u}du[\/latex].<\/li>\n
                5. Do not use integration by parts. Choose [latex]u[\/latex] to be [latex]u=3{x}^{3}+2[\/latex] and the integral can be put into the form [latex]\\displaystyle\\int \\sin\\left(u\\right)du[\/latex].<\/li>\n
                6. \n

                  The area under graph is [latex]0.39535[\/latex].
                  \n<\/span><\/p>\n


                  \n\"This<\/span><\/li>\n

                7. [latex]2\\pi e[\/latex]<\/li>\n
                8. [latex]2.05[\/latex]<\/li>\n
                9. [latex]12\\pi[\/latex]<\/li>\n
                10. [latex]8{\\pi }^{2}[\/latex]<\/li>\n<\/ol>\n

                  Trigonometric Integrals<\/h2>\n
                    \n
                  1. \n
                      \n
                    1. [latex]\\dfrac{{\\sin}^{4}x}{4}+C[\/latex]<\/li>\n
                    2. [latex]\\dfrac{1}{12}{\\tan}^{6}\\left(2x\\right)+C[\/latex]<\/li>\n
                    3. [latex]{\\sec}^{2}\\left(\\dfrac{x}{2}\\right)+C[\/latex]<\/li>\n
                    4. [latex]\\text{-}\\cos{x}+\\dfrac{1}{3}\\cos^{2}x+C[\/latex]<\/li>\n
                    5. [latex]-\\dfrac{1}{2}{\\cos}^{2}x+C[\/latex] or [latex]\\dfrac{1}{2}{\\sin}^{2}x+C[\/latex]<\/li>\n
                    6. [latex]-\\dfrac{1}{3}\\cos^{3}x+\\dfrac{2}{5}\\cos^{5}x-\\dfrac{1}{7}\\cos^{7}x+C[\/latex]<\/li>\n
                    7. [latex]\\dfrac{2}{3}{\\left(\\sin{x}\\right)}^{\\dfrac{3}{2}}+C[\/latex]<\/li>\n
                    8. [latex]\\sec{x}+C[\/latex]<\/li>\n
                    9. [latex]\\dfrac{1}{2}\\sec{x}\\tan{x}-\\dfrac{1}{2}\\text{ln}\\left(\\sec{x}+\\tan{x}\\right)+C[\/latex]<\/li>\n
                    10. [latex]\\dfrac{2\\tan{x}}{3}+\\dfrac{1}{3}\\sec{\\left(x\\right)}^{2}\\tan{x}[\/latex] [latex]=\\tan{x}+\\dfrac{{\\tan}^{3}x}{3}+C[\/latex]<\/li>\n
                    11. [latex]\\text{-}\\text{ln}|\\cot{x}+\\csc{x}|+C[\/latex]<\/li>\n
                    12. [latex]\\dfrac{{\\sin}^{3}\\left(ax\\right)}{3a}+C[\/latex]<\/li>\n
                    13. [latex]\\dfrac{\\pi }{2}[\/latex]<\/li>\n
                    14. [latex]\\dfrac{x}{2}+\\dfrac{1}{12}\\sin\\left(6x\\right)+C[\/latex]<\/li>\n
                    15. [latex]x+C[\/latex]<\/li>\n
                    16. [latex]0[\/latex]<\/li>\n
                    17. [latex]0[\/latex]<\/li>\n
                    18. [latex]0[\/latex]<\/li>\n
                    19. [latex]\\text{Approximately 0.239}[\/latex]<\/li>\n
                    20. [latex]\\sqrt{2}[\/latex]<\/li>\n
                    21. [latex]1.0[\/latex]<\/li>\n
                    22. [latex]0[\/latex]<\/li>\n
                    23. [latex]\\dfrac{3\\theta }{8}-\\dfrac{1}{4\\pi }\\sin\\left(2\\pi \\theta \\right)+\\dfrac{1}{32\\pi }\\sin\\left(4\\pi \\theta \\right)+C=f\\left(x\\right)[\/latex]<\/li>\n
                    24. [latex]\\text{ln}\\left(\\sqrt{3}\\right)[\/latex]<\/li>\n
                    25. [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }\\sin\\left(2x\\right)\\cos\\left(3x\\right)dx=0[\/latex]<\/li>\n
                    26. [latex]\\sqrt{\\tan\\left(x\\right)}x\\left(\\dfrac{8\\tan{x}}{21}+\\dfrac{2}{7}\\sec{x}^{2}\\tan{x}\\right)+C=f\\left(x\\right)[\/latex]<\/li>\n
                    27. The second integral is more difficult because the first integral is simply a u-substitution type.<\/li>\n
                    28. [latex]9{\\tan}^{2}\\theta[\/latex]<\/li>\n
                    29. [latex]{a}^{2}{\\text{cosh}}^{2}\\theta[\/latex]<\/li>\n
                    30. [latex]4{\\left(x-\\dfrac{1}{2}\\right)}^{2}[\/latex]<\/li>\n
                    31. [latex]\\text{-}{\\left(x+1\\right)}^{2}+5[\/latex]<\/li>\n
                    32. [latex]\\text{ln}|x+\\sqrt{\\text{-}{a}^{2}+{x}^{2}}|+C[\/latex]<\/li>\n
                    33. [latex]\\dfrac{1}{3}\\text{ln}|\\sqrt{9{x}^{2}+1}+3x|+C[\/latex]<\/li>\n
                    34. [latex]-\\dfrac{\\sqrt{1-{x}^{2}}}{x}+C[\/latex]<\/li>\n
                    35. [latex]9\\left[\\dfrac{x\\sqrt{{x}^{2}+9}}{18}+\\dfrac{1}{2}ln|\\dfrac{\\sqrt{{x}^{2}+9}}{3}+\\dfrac{x}{3}|\\right]+C[\/latex]<\/li>\n
                    36. [latex]-\\dfrac{1}{3}\\sqrt{9-{\\theta }^{2}}\\left(18+{\\theta }^{2}\\right)+C[\/latex]<\/li>\n
                    37. [latex]-\\dfrac{\\sqrt{1+{x}^{2}}}{x}+C[\/latex]<\/li>\n
                    38. [latex]\\dfrac{1}{8}\\left(x\\left(5 - 2{x}^{2}\\right)\\sqrt{1-{x}^{2}}+3\\text{arcsin}x\\right)+C[\/latex]<\/li>\n
                    39. [latex]\\text{ln}x-\\text{ln}|1+\\sqrt{1-{x}^{2}}|+C[\/latex]<\/li>\n
                    40. [latex]-\\dfrac{\\sqrt{-1+{x}^{2}}}{x}+\\text{ln}|x+\\sqrt{-1+{x}^{2}}|+C[\/latex]<\/li>\n
                    41. [latex]-\\dfrac{\\sqrt{1+{x}^{2}}}{x}+\\text{arcsinh}x+C[\/latex]<\/li>\n
                    42. [latex]-\\dfrac{1}{1+x}+C[\/latex]<\/li>\n
                    43. [latex]\\dfrac{2\\sqrt{-10+x}\\sqrt{x}\\text{ln}|\\sqrt{-10+x}+\\sqrt{x}|}{\\sqrt{\\left(10-x\\right)x}}+C[\/latex]<\/li>\n
                    44. [latex]\\dfrac{9\\pi }{2}[\/latex]; area of a semicircle with radius 3<\/li>\n
                    45. [latex]\\text{arcsin}\\left(x\\right)+C[\/latex] is the common answer.<\/li>\n
                    46. [latex]\\dfrac{1}{2}\\text{ln}\\left(1+{x}^{2}\\right)+C[\/latex] is the result using either method.<\/li>\n
                    47. Use trigonometric substitution. Let [latex]x=\\sec\\left(\\theta \\right)[\/latex].<\/li>\n
                    48. [latex]4.367[\/latex]<\/li>\n
                    49. [latex]\\dfrac{{\\pi }^{2}}{8}+\\dfrac{\\pi }{4}[\/latex]<\/li>\n
                    50. [latex]y=\\dfrac{1}{16}\\text{ln}|\\dfrac{x+8}{x - 8}|+3[\/latex]<\/li>\n
                    51. [latex]24.6[\/latex] m\u00b3<\/li>\n
                    52. [latex]\\dfrac{2\\pi }{3}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                      Partial Fractions<\/h2>\n
                        \n
                      1. [latex]-\\dfrac{2}{x+1}+\\dfrac{5}{2\\left(x+2\\right)}+\\dfrac{1}{2x}[\/latex]<\/li>\n
                      2. [latex]\\dfrac{1}{{x}^{2}}+\\dfrac{3}{x}[\/latex]<\/li>\n
                      3. [latex]2{x}^{2}+4x+8+\\dfrac{16}{x - 2}[\/latex]<\/li>\n
                      4. [latex]-\\dfrac{1}{{x}^{2}}-\\dfrac{1}{x}+\\dfrac{1}{x - 1}[\/latex]<\/li>\n
                      5. [latex]-\\dfrac{1}{2\\left(x - 2\\right)}+\\dfrac{1}{2\\left(x - 1\\right)}-\\dfrac{1}{6x}+\\dfrac{1}{6\\left(x - 3\\right)}[\/latex]<\/li>\n
                      6. [latex]\\dfrac{1}{x - 1}+\\dfrac{2x+1}{{x}^{2}+x+1}[\/latex]<\/li>\n
                      7. [latex]\\dfrac{2}{x+1}+\\dfrac{x}{{x}^{2}+4}-\\dfrac{1}{{\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/li>\n
                      8. [latex]\\text{-}\\text{ln}|2-x|+2\\text{ln}|4+x|+C[\/latex]<\/li>\n
                      9. [latex]\\dfrac{1}{2}\\text{ln}|4-{x}^{2}|+C[\/latex]<\/li>\n
                      10. [latex]2\\left(x+\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{1+x}{3}\\right)\\right)+C[\/latex]<\/li>\n
                      11. [latex]2\\text{ln}|x|-3\\text{ln}|1+x|+C[\/latex]<\/li>\n
                      12. [latex]\\dfrac{1}{16}\\left(\\text{-}\\dfrac{4}{-2+x}-\\text{ln}|-2+x|+\\text{ln}|2+x|\\right)+C[\/latex]<\/li>\n
                      13. [latex]\\dfrac{1}{30}\\left(-2\\sqrt{5}\\text{arctan}\\left[\\dfrac{1+x}{\\sqrt{5}}\\right]+2\\text{ln}|-4+x|-\\text{ln}|6+2x+{x}^{2}|\\right)+C[\/latex]<\/li>\n
                      14. [latex]-\\dfrac{3}{x}+4\\text{ln}|x+2|+x+C[\/latex]<\/li>\n
                      15. [latex]\\text{-}\\text{ln}|3-x|+\\dfrac{1}{2}\\text{ln}|{x}^{2}+4|+C[\/latex]<\/li>\n
                      16. [latex]\\text{ln}|x - 2|-\\dfrac{1}{2}\\text{ln}|{x}^{2}+2x+2|+C[\/latex]<\/li>\n
                      17. [latex]\\text{-}x+\\text{ln}|1-{e}^{x}|+C[\/latex]<\/li>\n
                      18. [latex]\\dfrac{1}{5}\\text{ln}|\\dfrac{\\cos{x}+3}{\\cos{x} - 2}|+C[\/latex]<\/li>\n
                      19. [latex]\\dfrac{1}{2 - 2{e}^{2t}}+C[\/latex]<\/li>\n
                      20. [latex]2\\sqrt{1+x}-2\\text{ln}|1+\\sqrt{1+x}|+C[\/latex]<\/li>\n
                      21. [latex]\\text{ln}|\\dfrac{\\sin{x}}{1-\\sin{x}}|+C[\/latex]<\/li>\n
                      22. [latex]\\dfrac{\\sqrt{3}}{4}[\/latex]<\/li>\n
                      23. [latex]x-\\text{ln}\\left(1+{e}^{x}\\right)+C[\/latex]<\/li>\n
                      24. [latex]6{x}^{\\dfrac{1}{6}}-3{x}^{\\dfrac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left(1+{x}^{\\dfrac{1}{6}}\\right)+C[\/latex]<\/li>\n
                      25. [latex]\\dfrac{4}{3}\\pi \\text{arctanh}\\left[\\dfrac{1}{3}\\right]=\\dfrac{1}{3}\\pi \\text{ln}4[\/latex]<\/li>\n
                      26. [latex]x=\\text{-}\\text{ln}|t - 3|+\\text{ln}|t - 4|+\\text{ln}2[\/latex]<\/li>\n
                      27. [latex]x=\\text{ln}|t - 1|-\\sqrt{2}\\text{arctan}\\left(\\sqrt{2}t\\right)-\\dfrac{1}{2}\\text{ln}\\left({t}^{2}+\\dfrac{1}{2}\\right)+\\sqrt{2}\\text{arctan}\\left(2\\sqrt{2}\\right)+\\dfrac{1}{2}\\text{ln}4.5[\/latex]<\/li>\n
                      28. [latex]\\dfrac{2}{5}\\pi \\text{ln}\\dfrac{28}{13}[\/latex]<\/li>\n
                      29. [latex]\\dfrac{\\text{arctan}\\left[\\dfrac{-1+2x}{\\sqrt{3}}\\right]}{\\sqrt{3}}+\\dfrac{1}{3}\\text{ln}|1+x|-\\dfrac{1}{6}\\text{ln}|1-x+{x}^{2}|+C[\/latex]<\/li>\n
                      30. [latex]2.0[\/latex] in.\u00b2<\/li>\n
                      31. [latex]3{\\left(-8+x\\right)}^{\\dfrac{1}{3}}[\/latex] [latex]-2\\sqrt{3}\\text{arctan}\\left[\\dfrac{-1+{\\left(-8+x\\right)}^{\\dfrac{1}{3}}}{\\sqrt{3}}\\right][\/latex] [latex]-2\\text{ln}\\left[2+{\\left(-8+x\\right)}^{\\dfrac{1}{3}}\\right][\/latex] [latex]+\\text{ln}\\left[4 - 2{\\left(-8+x\\right)}^{\\dfrac{1}{3}}+{\\left(-8+x\\right)}^{\\dfrac{2}{3}}\\right]+C[\/latex]<\/li>\n<\/ol>\n

                        Other Strategies for Integration<\/h2>\n
                          \n
                        1. [latex]\\dfrac{1}{2}\\text{ln}|{x}^{2}+2x+2|+2\\text{arctan}\\left(x+1\\right)+C[\/latex]<\/li>\n
                        2. [latex]{\\text{cosh}}^{-1}\\left(\\dfrac{x+3}{3}\\right)+C[\/latex]<\/li>\n
                        3. [latex]\\dfrac{{2}^{{x}^{2}-1}}{\\text{ln}2}+C[\/latex]<\/li>\n
                        4. [latex]\\text{arcsin}\\left(\\dfrac{y}{2}\\right)+C[\/latex]<\/li>\n
                        5. [latex]-\\dfrac{1}{2}\\csc\\left(2w\\right)+C[\/latex]<\/li>\n
                        6. [latex]9 - 6\\sqrt{2}[\/latex]<\/li>\n
                        7. [latex]2-\\dfrac{\\pi }{2}[\/latex]<\/li>\n
                        8. [latex]\\dfrac{1}{12}{\\tan}^{4}\\left(3x\\right)-\\dfrac{1}{6}{\\tan}^{2}\\left(3x\\right)+\\dfrac{1}{3}\\text{ln}|\\sec\\left(3x\\right)|+C[\/latex]<\/li>\n
                        9. [latex]2\\cot\\left(\\dfrac{w}{2}\\right)-2\\csc\\left(\\dfrac{w}{2}\\right)+w+C[\/latex]<\/li>\n
                        10. [latex]\\dfrac{1}{5}\\text{ln}|\\dfrac{2\\left(5+4\\sin{t} - 3\\cos{t}\\right)}{4\\cos{t}+3\\sin{t}}|[\/latex]<\/li>\n
                        11. [latex]6{x}^{\\dfrac{1}{6}}-3{x}^{\\dfrac{1}{3}}+2\\sqrt{x}-6\\text{ln}\\left[1+{x}^{\\dfrac{1}{6}}\\right]+C[\/latex]<\/li>\n
                        12. [latex]\\text{-}{x}^{3}\\cos{x}+3{x}^{2}\\sin{x}+6x\\cos{x} - 6\\sin{x}+C[\/latex]<\/li>\n
                        13. [latex]\\dfrac{1}{2}\\left({x}^{2}+\\text{ln}|1+{e}^{\\text{-}{x}^{2}}|\\right)+C[\/latex]<\/li>\n
                        14. [latex]2\\text{arctan}\\left(\\sqrt{x - 1}\\right)+C[\/latex]<\/li>\n
                        15. [latex]0.5=\\dfrac{1}{2}[\/latex]<\/li>\n
                        16. [latex]8.0[\/latex]<\/li>\n
                        17. [latex]\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{1}{3}\\left(x+2\\right)\\right)+C[\/latex]<\/li>\n
                        18. [latex]\\dfrac{1}{3}\\text{arctan}\\left(\\dfrac{x+1}{3}\\right)+C[\/latex]<\/li>\n
                        19. [latex]\\text{ln}\\left({e}^{x}+\\sqrt{4+{e}^{2x}}\\right)+C[\/latex]<\/li>\n
                        20. [latex]\\text{ln}x-\\dfrac{1}{6}\\text{ln}\\left({x}^{6}+1\\right)-\\dfrac{\\text{arctan}\\left({x}^{3}\\right)}{3{x}^{3}}+C[\/latex]<\/li>\n
                        21. [latex]\\text{ln}|x+\\sqrt{16+{x}^{2}}|+C[\/latex]<\/li>\n
                        22. [latex]-\\dfrac{1}{4}\\cot\\left(2x\\right)+C[\/latex]<\/li>\n
                        23. [latex]\\dfrac{1}{2}\\text{arctan}10[\/latex]<\/li>\n
                        24. [latex]1276.14[\/latex]<\/li>\n
                        25. [latex]7.21[\/latex]<\/li>\n
                        26. [latex]\\sqrt{5}-\\sqrt{2}+\\text{ln}|\\dfrac{2+2\\sqrt{2}}{1+\\sqrt{5}}|[\/latex]<\/li>\n
                        27. [latex]\\dfrac{1}{3}\\text{arctan}\\left(3\\right)\\approx 0.416[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":6,"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\/179"}],"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\/179\/revisions"}],"predecessor-version":[{"id":191,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/179\/revisions\/191"}],"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\/179\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=179"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=179"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=179"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}