{"id":737,"date":"2025-06-20T17:08:33","date_gmt":"2025-06-20T17:08:33","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&p=737"},"modified":"2025-08-13T15:27:45","modified_gmt":"2025-08-13T15:27:45","slug":"advanced-integration-techniques-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/advanced-integration-techniques-get-stronger\/","title":{"raw":"Advanced Integration Techniques: Get Stronger","rendered":"Advanced Integration Techniques: Get Stronger"},"content":{"raw":"

Integration by Parts<\/h2>\r\nIn the following exercises (1-3), use the guidelines in this section to choose u.<\/em> Do not <\/em>evaluate the integrals.<\/strong>\r\n
    \r\n \t
  1. [latex]\\displaystyle\\int {x}^{3}{e}^{2x}dx[\/latex]<\/li>\r\n \t
  2. \u00a0<\/strong>[latex]\\displaystyle\\int {y}^{3}\\cos{y} dy[\/latex]<\/li>\r\n \t
  3. [latex]\\displaystyle\\int {e}^{3x}\\sin\\left(2x\\right)dx[\/latex]<\/span><\/li>\r\n<\/ol>\r\nFor the following exercises (4-19), find\u00a0the integral by using the simplest method. Not all problems require integration by parts.<\/em><\/strong>\r\n
      \r\n \t
    1. [latex]\\displaystyle\\int \\text{ln}xdx[\/latex] (Hint:<\/em> [latex]\\displaystyle\\int \\text{ln}xdx[\/latex] is equivalent to [latex]\\displaystyle\\int 1\\cdot \\text{ln}\\left( x\\right)dx.[\/latex])<\/li>\r\n \t
    2. [latex]\\displaystyle\\int {\\tan}^{-1}xdx[\/latex]<\/li>\r\n \t
    3. \u00a0<\/strong>[latex]\\displaystyle\\int x\\sin\\left(2x\\right)dx[\/latex]<\/span><\/li>\r\n \t
    4. [latex]\\displaystyle\\int x{e}^{\\text{-}x}dx[\/latex]<\/li>\r\n \t
    5. [latex]\\displaystyle\\int {x}^{2}\\cos{x}dx[\/latex]<\/li>\r\n \t
    6. \u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}\\left(2x+1\\right)dx[\/latex]<\/li>\r\n \t
    7. [latex]\\displaystyle\\int {e}^{x}\\sin{x}dx[\/latex]<\/li>\r\n \t
    8. [latex]\\displaystyle\\int x{e}^{\\text{-}{x}^{2}}dx[\/latex]<\/li>\r\n \t
    9. \u00a0<\/strong>[latex]\\displaystyle\\int \\sin\\left(\\text{ln}\\left(2x\\right)\\right)dx[\/latex]<\/li>\r\n \t
    10. [latex]\\displaystyle\\int {\\left(\\text{ln}x\\right)}^{2}dx[\/latex]<\/li>\r\n \t
    11. \u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\text{ln}xdx[\/latex]<\/li>\r\n \t
    12. \u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{-1}\\left(2x\\right)dx[\/latex]<\/li>\r\n \t
    13. [latex]\\displaystyle\\int {x}^{2}\\sin{x}dx[\/latex]<\/li>\r\n \t
    14. [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/li>\r\n \t
    15. [latex]\\displaystyle\\int x{\\sec}^{-1}xdx[\/latex]<\/li>\r\n \t
    16. [latex]\\displaystyle\\int x\\text{cosh}xdx[\/latex]<\/span><\/li>\r\n<\/ol>\r\nFor the following exercises (20-24), compute the definite integrals. Use a graphing utility to confirm your answers.<\/strong>\r\n
        \r\n \t
      1. \u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}x{e}^{-2x}dx[\/latex] (Express the answer in exact form.)<\/li>\r\n \t
      2. [latex]{\\displaystyle\\int }_{1}^{e}\\text{ln}\\left({x}^{2}\\right)dx[\/latex]<\/span><\/li>\r\n \t
      3. [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }x\\sin{x}dx[\/latex] (Express the answer in exact form.)<\/li>\r\n \t
      4. \u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{x}^{2}\\sin{x}dx[\/latex] (Express the answer in exact form.)<\/li>\r\n \t
      5. Evaluate [latex]\\displaystyle\\int \\cos{x}\\text{ln}\\left(\\sin{x}\\right)dx[\/latex]<\/li>\r\n<\/ol>\r\nDerive the following formulas (25-26) using the technique of integration by parts. Assume that <\/em><\/span>[latex]n[\/latex] is a positive integer. These formulas are called <\/span>reduction formulas because the exponent in the <\/span>[latex]x[\/latex] term has been reduced by one in each case. The second integral is simpler than the original integral.<\/span><\/em><\/strong>\r\n
          \r\n \t
        1. [latex]\\displaystyle\\int {x}^{n}\\cos{x}dx={x}^{n}\\sin{x}-n\\displaystyle\\int {x}^{n - 1}\\sin{x}dx[\/latex]<\/li>\r\n \t
        2. \r\n

          Integrate [latex]\\displaystyle\\int 2x\\sqrt{2x - 3}dx[\/latex] using two methods:<\/p>\r\n\r\n

            \r\n \t
          1. Using parts, letting [latex]dv=\\sqrt{2x - 3}dx[\/latex]<\/li>\r\n \t
          2. Substitution, letting [latex]u=2x - 3[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nIn the following exercises (27-29), state whether you would use integration by parts to evaluate the integral. If so, identify [latex]u [\/latex]and [latex]dv.[\/latex] If not, describe the technique used to perform the integration without actually doing the problem.<\/strong>\r\n
              \r\n \t
            1. [latex]\\displaystyle\\int \\frac{{\\text{ln}}^{2}x}{x}dx[\/latex]<\/li>\r\n \t
            2. \u00a0<\/strong>[latex]\\displaystyle\\int x{e}^{{x}^{2}-3}dx[\/latex]<\/span><\/li>\r\n \t
            3. \u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\sin\\left(3{x}^{3}+2\\right)dx[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercise, sketch the region bounded above by the curve, the [latex]x[\/latex]-axis, and [latex]x=1[\/latex], and find the area of the region. Provide the exact form or round answers to the number of places indicated.<\/strong>\r\n
                \r\n \t
              1. [latex]y={e}^{\\text{-}x}\\sin\\left(\\pi x\\right)[\/latex] (Approximate answer to five decimal places.)<\/li>\r\n<\/ol>\r\nFind the volume generated by rotating the region bounded by the given curves about the specified line for the following exercises (31-34). Express the answers in exact form or approximate to the number of decimal places indicated.<\/strong>\r\n
                  \r\n \t
                1. [latex]y={e}^{\\text{-}x}[\/latex] [latex]y=0,x=-1x=0[\/latex]; about [latex]x=1[\/latex] (Express the answer in exact form.)<\/li>\r\n \t
                2. Find the area under the graph of [latex]y={\\sec}^{3}x[\/latex] from [latex]x=0\\text{to}x=1[\/latex]. (Round the answer to two significant digits.)<\/li>\r\n \t
                3. \r\n

                  Find the area of the region enclosed by the curve [latex]y=x\\cos{x}[\/latex] and the x<\/em>-axis for [latex]\\frac{11\\pi }{2}\\le x\\le \\frac{13\\pi }{2}[\/latex]. (Express the answer in exact form.)<\/p>\r\n<\/li>\r\n \t

                4. Find the volume of the solid generated by revolving the region bounded by the curve [latex]y=4\\cos{x}[\/latex] and the x<\/em>-axis, [latex]\\frac{\\pi }{2}\\le x\\le \\frac{3\\pi }{2}[\/latex], about the x<\/em>-axis. (Express the answer in exact form.)<\/li>\r\n<\/ol>\r\n

                  Trigonometric Integrals<\/h2>\r\n

                  For the following exercises (1-3), evaluate each of the following integrals by u<\/em>-substitution.<\/strong><\/p>\r\n\r\n

                    \r\n \t
                  1. [latex]\\displaystyle\\int {\\sin}^{3}x\\cos{x}dx[\/latex]<\/li>\r\n \t
                  2. [latex]\\displaystyle\\int {\\tan}^{5}\\left(2x\\right){\\sec}^{2}\\left(2x\\right)dx[\/latex]<\/li>\r\n \t
                  3. [latex]\\displaystyle\\int \\tan\\left(\\dfrac{x}{2}\\right){\\sec}^{2}\\left(\\dfrac{x}{2}\\right)dx[\/latex]<\/li>\r\n<\/ol>\r\n

                    In the following exercises (4-11), compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions.<\/strong><\/p>\r\n\r\n

                      \r\n \t
                    1. [latex]\\displaystyle\\int {\\sin}^{3}xdx[\/latex]<\/li>\r\n \t
                    2. [latex]\\displaystyle\\int \\sin{x}\\cos{x}dx[\/latex]<\/li>\r\n \t
                    3. [latex]\\displaystyle\\int {\\sin}^{5}x{\\cos}^{2}xdx[\/latex]<\/li>\r\n \t
                    4. [latex]\\displaystyle\\int \\sqrt{\\sin{x}}\\cos{x}dx[\/latex]<\/li>\r\n \t
                    5. [latex]\\displaystyle\\int \\sec{x}\\tan{x}dx[\/latex]<\/li>\r\n \t
                    6. [latex]\\displaystyle\\int {\\tan}^{2}x\\sec{x}dx[\/latex]<\/li>\r\n \t
                    7. [latex]\\displaystyle\\int {\\sec}^{4}xdx[\/latex]<\/li>\r\n \t
                    8. [latex]\\displaystyle\\int \\csc{x}dx[\/latex]<\/li>\r\n<\/ol>\r\n

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

                        \r\n \t
                      1. [latex]\\displaystyle\\int {\\sin}^{2}ax\\cos{ax} dx[\/latex]<\/li>\r\n<\/ol>\r\n

                        Use the double-angle formulas to evaluate the following integrals (13-15).<\/strong><\/p>\r\n\r\n

                          \r\n \t
                        1. [latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}xdx[\/latex]<\/li>\r\n \t
                        2. [latex]\\displaystyle\\int {\\cos}^{2}3xdx[\/latex]<\/li>\r\n \t
                        3. [latex]\\displaystyle\\int {\\sin}^{2}xdx+\\displaystyle\\int {\\cos}^{2}xdx[\/latex]<\/li>\r\n<\/ol>\r\n

                          For the following exercises (16-22), evaluate the definite integrals. Express answers in exact form whenever possible.<\/strong><\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]{\\displaystyle\\int }_{0}^{2\\pi }\\cos{x}\\sin2xdx[\/latex]<\/li>\r\n \t
                          2. [latex]{\\displaystyle\\int }_{0}^{\\pi }\\cos\\left(99x\\right)\\sin\\left(101x\\right)dx[\/latex]<\/li>\r\n \t
                          3. [latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sin{x}\\sin\\left(2x\\right)\\sin\\left(3x\\right)dx[\/latex]<\/li>\r\n \t
                          4. [latex]{\\displaystyle\\int }_{\\dfrac{\\pi}{6}}^{\\dfrac{\\pi}{3}}\\dfrac{{\\cos}^{3}x}{\\sqrt{\\sin{x}}}dx[\/latex] (Round this answer to three decimal places.)<\/li>\r\n \t
                          5. [latex]{\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\sqrt{1-\\cos\\left(2x\\right)}dx[\/latex]<\/li>\r\n \t
                          6. Find the area of the region bounded by the graphs of the equations [latex]y={\\cos}^{2}x,y={\\sin}^{2}x,x=-\\dfrac{\\pi }{4},\\text{and }x=\\dfrac{\\pi }{4}[\/latex].<\/li>\r\n \t
                          7. 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].<\/li>\r\n<\/ol>\r\n

                            For the following exercises (23-24), solve the differential equations.<\/strong><\/p>\r\n\r\n

                              \r\n \t
                            1. [latex]\\dfrac{dy}{d\\theta }={\\sin}^{4}\\left(\\pi \\theta \\right)[\/latex]<\/li>\r\n \t
                            2. Find the length of the curve [latex]y=\\text{ln}\\left(\\sin{x}\\right),\\dfrac{\\pi }{3}\\le x\\le \\dfrac{\\pi }{2}[\/latex].<\/li>\r\n<\/ol>\r\n

                              For the following exercises (25-26), use this information: <\/strong><\/p>\r\n\r\n

                                \r\n \t
                              • The inner product of two functions [latex]f[\/latex] and [latex]g[\/latex] over [latex]\\left[a,b\\right][\/latex] is defined by<\/strong><\/strong>
                                [latex]f\\left(x\\right)\\cdot g\\left(x\\right)=\\langle f,g\\rangle ={\\displaystyle\\int }_{a}^{b}f\\cdot gdx[\/latex]<\/center><\/li>\r\n \t
                              • Two distinct functions [latex]f[\/latex] and [latex]g[\/latex] are said to be orthogonal if [latex]\\langle f,g\\rangle =0[\/latex].<\/strong><\/li>\r\n<\/ul>\r\n
                                  \r\n \t
                                1. Show that [latex]{\\sin(2x),\\cos(3x)}[\/latex] are orthogonal over the interval [latex][\\text{-}\\pi ,\\pi ][\/latex].<\/li>\r\n \t
                                2. Integrate [latex]{y}^{\\prime }=\\sqrt{\\tan{x}}{\\sec}^{4}x[\/latex].<\/li>\r\n<\/ol>\r\n

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

                                    \r\n \t
                                  1. [latex]\\displaystyle\\int {\\tan}^{350}x{\\sec}^{2}xdx[\/latex] or [latex]\\displaystyle\\int {\\tan}^{350}x\\sec{x}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                    Simplify the following expressions by writing each one using a single trigonometric function.<\/strong><\/p>\r\n\r\n

                                      \r\n \t
                                    1. [latex]9{\\sec}^{2}\\theta -9[\/latex]<\/li>\r\n \t
                                    2. [latex]{a}^{2}+{a}^{2}{\\text{sinh}}^{2}\\theta [\/latex]<\/li>\r\n<\/ol>\r\n

                                      Use the technique of completing the square to express each trinomial as the square of a binomial.<\/strong><\/p>\r\n\r\n

                                        \r\n \t
                                      1. [latex]4{x}^{2}-4x+1[\/latex]<\/li>\r\n \t
                                      2. [latex]\\text{-}{x}^{2}-2x+4[\/latex]<\/li>\r\n<\/ol>\r\n

                                        In the following exercises (32-38), integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.<\/strong><\/p>\r\n\r\n

                                          \r\n \t
                                        1. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/li>\r\n \t
                                        2. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+9{x}^{2}}}[\/latex]<\/li>\r\n \t
                                        3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\r\n \t
                                        4. [latex]\\displaystyle\\int \\sqrt{{x}^{2}+9}dx[\/latex]<\/li>\r\n \t
                                        5. [latex]\\displaystyle\\int \\frac{{\\theta }^{3}d\\theta }{\\sqrt{9-{\\theta }^{2}}}d\\theta [\/latex]<\/li>\r\n \t
                                        6. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\r\n \t
                                        7. [latex]{\\displaystyle\\int }_{-1}^{1}{\\left(1-{x}^{2}\\right)}^{\\frac{3}{2}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                          In the following exercises (39-41), use the substitutions [latex]x=\\text{sinh}\\theta ,\\text{cosh}\\theta [\/latex], or [latex]\\text{tanh}\\theta [\/latex]. Express the final answers in terms of the variable [latex]x[\/latex].<\/strong><\/p>\r\n\r\n

                                            \r\n \t
                                          1. [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\r\n \t
                                          2. [latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-1}}{{x}^{2}}dx[\/latex]<\/li>\r\n \t
                                          3. [latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}}{{x}^{2}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                            Use the technique of completing the square to evaluate the following integrals (42-43).<\/strong><\/p>\r\n\r\n

                                              \r\n \t
                                            1. [latex]\\displaystyle\\int \\frac{1}{{x}^{2}+2x+1}dx[\/latex]<\/li>\r\n \t
                                            2. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+10x}}dx[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (44-49), solve each problem.<\/strong>\r\n
                                                \r\n \t
                                              1. Evaluate the integral without using calculus: [latex]{\\displaystyle\\int }_{-3}^{3}\\sqrt{9-{x}^{2}}dx[\/latex].<\/li>\r\n \t
                                              2. Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex] using two different substitutions. First, let [latex]x=\\cos\\theta [\/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\\sin\\theta [\/latex] and use trigonometric substitution. Are the answers the same?<\/li>\r\n \t
                                              3. Evaluate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+1}dx[\/latex] using the form [latex]\\displaystyle\\int \\frac{1}{u}du[\/latex]. Next, evaluate the same integral using [latex]x=\\tan\\theta [\/latex]. Are the results the same?<\/li>\r\n \t
                                              4. State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int {x}^{2}\\sqrt{{x}^{2}-1}dx[\/latex]. Why did you choose this method?<\/li>\r\n \t
                                              5. Find the length of the arc of the curve over the specified interval: [latex]y=\\text{ln}x,\\left[1,5\\right][\/latex]. Round the answer to three decimal places.<\/li>\r\n \t
                                              6. The region bounded by the graph of [latex]f\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex] and the [latex]x[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex] is revolved about the [latex]x[\/latex]-axis. Find the volume of the solid that is generated.<\/li>\r\n<\/ol>\r\n

                                                For the following exercises (50-52), solve the initial-value problem for [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\r\n\r\n

                                                  \r\n \t
                                                1. [latex]\\left(64-{x}^{2}\\right)\\frac{dy}{dx}=1,y\\left(0\\right)=3[\/latex]<\/li>\r\n \t
                                                2. An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\\frac{16}{\\sqrt{64+{x}^{2}}},x=0,y=0,x=2[\/latex] about the [latex]x[\/latex]-axis. Find the volume of the tank (in cubic meters).<\/li>\r\n \t
                                                3. Find the length of the curve [latex]y=\\sqrt{16-{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=2[\/latex].<\/li>\r\n<\/ol>\r\n

                                                  Partial Fractions<\/h2>\r\n

                                                  In the following exercises (1-7), express the rational function as a sum or difference of two simpler rational expressions.<\/strong><\/p>\r\n\r\n

                                                    \r\n \t
                                                  1. [latex]\\frac{{x}^{2}+1}{x\\left(x+1\\right)\\left(x+2\\right)}[\/latex]<\/li>\r\n \t
                                                  2. [latex]\\frac{3x+1}{{x}^{2}}[\/latex]<\/li>\r\n \t
                                                  3. [latex]\\frac{2{x}^{4}}{{x}^{2}-2x}[\/latex]<\/li>\r\n \t
                                                  4. [latex]\\frac{1}{{x}^{2}\\left(x - 1\\right)}[\/latex]<\/li>\r\n \t
                                                  5. [latex]\\frac{1}{x\\left(x - 1\\right)\\left(x - 2\\right)\\left(x - 3\\right)}[\/latex]<\/li>\r\n \t
                                                  6. [latex]\\frac{3{x}^{2}}{{x}^{3}-1}=\\frac{3{x}^{2}}{\\left(x - 1\\right)\\left({x}^{2}+x+1\\right)}[\/latex]<\/li>\r\n \t
                                                  7. [latex]\\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\\left(x+1\\right){\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/li>\r\n<\/ol>\r\n

                                                    Use the method of partial fractions to evaluate each of the following integrals (8-12).<\/strong><\/p>\r\n\r\n

                                                      \r\n \t
                                                    1. [latex]\\displaystyle\\int \\frac{3x}{{x}^{2}+2x - 8}dx[\/latex]<\/li>\r\n \t
                                                    2. [latex]\\displaystyle\\int \\frac{x}{{x}^{2}-4}dx[\/latex]<\/li>\r\n \t
                                                    3. [latex]\\displaystyle\\int \\frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[\/latex]<\/li>\r\n \t
                                                    4. [latex]\\displaystyle\\int \\frac{2-x}{{x}^{2}+x}dx[\/latex]<\/li>\r\n \t
                                                    5. [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[\/latex]<\/li>\r\n<\/ol>\r\n

                                                      Evaluate the following integrals (13-14), which have irreducible quadratic factors.<\/strong><\/p>\r\n\r\n

                                                        \r\n \t
                                                      1. [latex]\\displaystyle\\int \\frac{2}{\\left(x - 4\\right)\\left({x}^{2}+2x+6\\right)}dx[\/latex]<\/li>\r\n \t
                                                      2. [latex]\\displaystyle\\int \\frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                                        Use the method of partial fractions to evaluate the following integrals (15-16).<\/strong><\/p>\r\n\r\n

                                                          \r\n \t
                                                        1. [latex]\\displaystyle\\int \\frac{3x+4}{\\left({x}^{2}+4\\right)\\left(3-x\\right)}dx[\/latex]<\/li>\r\n \t
                                                        2. [latex]\\displaystyle\\int \\frac{3x+4}{{x}^{3}-2x - 4}dx[\/latex] (Hint: Use the rational root theorem.)<\/li>\r\n<\/ol>\r\n

                                                          Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.<\/strong><\/p>\r\n\r\n

                                                            \r\n \t
                                                          1. [latex]\\displaystyle\\int \\frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[\/latex]<\/li>\r\n \t
                                                          2. [latex]\\displaystyle\\int \\frac{\\sin{x}}{{\\cos}^{2}x+\\cos{x} - 6}dx[\/latex]<\/li>\r\n \t
                                                          3. [latex]\\displaystyle\\int \\frac{dt}{{\\left({e}^{t}-{e}^{\\text{-}t}\\right)}^{2}}[\/latex]<\/li>\r\n \t
                                                          4. [latex]\\displaystyle\\int \\frac{dx}{1+\\sqrt{x+1}}[\/latex]<\/li>\r\n \t
                                                          5. [latex]\\displaystyle\\int \\frac{\\cos{x}}{\\sin{x}\\left(1-\\sin{x}\\right)}dx[\/latex]<\/li>\r\n \t
                                                          6. [latex]\\underset{1}{\\overset{2}{\\displaystyle\\int }}\\frac{1}{{x}^{2}\\sqrt{4-{x}^{2}}}dx[\/latex]<\/li>\r\n \t
                                                          7. [latex]\\displaystyle\\int \\frac{1}{1+{e}^{x}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                                            For the following exercises (44-49), use the given substitution to convert the integral to an integral of a rational function, then evaluate.<\/strong><\/p>\r\n\r\n

                                                              \r\n \t
                                                            1. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{x}+\\sqrt[3]{x}}dx;x={u}^{6}[\/latex]<\/li>\r\n \t
                                                            2. Find the volume of the solid generated when the region bounded by [latex]y=\\frac{1}{\\sqrt{x\\left(3-x\\right)}}[\/latex], [latex]y=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex] is revolved about the x-axis.<\/li>\r\n<\/ol>\r\n

                                                              In the following exercises (1-7), solve the initial-value problem for [latex]x[\/latex] as a function of [latex]t[\/latex].<\/strong><\/p>\r\n\r\n

                                                                \r\n \t
                                                              1. [latex]\\left({t}^{2}-7t+12\\right)\\frac{dx}{dt}=1,\\left(t>4,x\\left(5\\right)=0\\right)[\/latex]<\/li>\r\n \t
                                                              2. [latex]\\left(2{t}^{3}-2{t}^{2}+t - 1\\right)\\frac{dx}{dt}=3,x\\left(2\\right)=0[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (28-31), solve each problem.<\/strong>\r\n
                                                                  \r\n \t
                                                                1. Find the volume generated by revolving the area bounded by [latex]y=\\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\\text{and }y=0[\/latex] about the y-axis.<\/li>\r\n \t
                                                                2. Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}+1}[\/latex].<\/li>\r\n \t
                                                                3. Find the area under the curve [latex]y=\\frac{1}{1+\\sin{x}}[\/latex] between [latex]x=0[\/latex] and [latex]x=\\pi [\/latex]. (Assume the dimensions are in inches.)<\/li>\r\n \t
                                                                4. Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt[3]{x - 8}}{x}dx[\/latex].<\/li>\r\n<\/ol>\r\n

                                                                  Other Strategies for Integration<\/h2>\r\n

                                                                  For the following exercises (1-8), use a table of integrals to evaluate the following integrals.<\/strong><\/p>\r\n\r\n

                                                                    \r\n \t
                                                                  1. [latex]\\displaystyle\\int \\frac{x+3}{{x}^{2}+2x+2}dx[\/latex]<\/li>\r\n \t
                                                                  2. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{{x}^{2}+6x}}dx[\/latex]<\/li>\r\n \t
                                                                  3. [latex]\\displaystyle\\int x\\cdot {2}^{{x}^{2}}dx[\/latex]<\/li>\r\n \t
                                                                  4. [latex]\\displaystyle\\int \\frac{dy}{\\sqrt{4-{y}^{2}}}[\/latex]<\/li>\r\n \t
                                                                  5. [latex]\\displaystyle\\int \\csc\\left(2w\\right)\\cot\\left(2w\\right)dw[\/latex]<\/li>\r\n \t
                                                                  6. [latex]{\\displaystyle\\int }_{0}^{1}\\frac{3xdx}{\\sqrt{{x}^{2}+8}}[\/latex]<\/li>\r\n \t
                                                                  7. [latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\tan}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/li>\r\n \t
                                                                  8. [latex]\\displaystyle\\int {\\tan}^{5}\\left(3x\\right)dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                                                    Use a CAS to evaluate the following integrals (9-14). Tables can also be used to verify the answers.<\/strong><\/p>\r\n\r\n

                                                                      \r\n \t
                                                                    1. [latex]\\displaystyle\\int \\frac{dw}{1+\\sec\\left(\\frac{w}{2}\\right)}[\/latex]<\/li>\r\n \t
                                                                    2. [latex]{\\displaystyle\\int }_{0}^{t}\\frac{dt}{4\\cos{t}+3\\sin{t}}[\/latex]<\/li>\r\n \t
                                                                    3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{\\frac{1}{2}}+{x}^{\\frac{1}{3}}}[\/latex]<\/li>\r\n \t
                                                                    4. [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/li>\r\n \t
                                                                    5. [latex]\\displaystyle\\int \\frac{x}{1+{e}^{\\text{-}{x}^{2}}}dx[\/latex]<\/li>\r\n \t
                                                                    6. [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/li>\r\n<\/ol>\r\n

                                                                      Use a calculator or CAS to evaluate the following integrals (15-17).<\/strong><\/p>\r\n\r\n

                                                                        \r\n \t
                                                                      1. [latex]{\\displaystyle\\int }_{0}^{\\frac{z\\pi}{4}}\\cos\\left(2x\\right)dx[\/latex]<\/li>\r\n \t
                                                                      2. [latex]{\\displaystyle\\int }_{0}^{8}\\frac{2x}{\\sqrt{{x}^{2}+36}}dx[\/latex]<\/li>\r\n \t
                                                                      3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+4x+13}[\/latex]<\/li>\r\n<\/ol>\r\n

                                                                        Use tables to evaluate the following integrals (18-20). You may need to complete the square or change variables to put the integral into a form given in the table.<\/strong><\/p>\r\n\r\n

                                                                          \r\n \t
                                                                        1. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+2x+10}[\/latex]<\/li>\r\n \t
                                                                        2. [latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-4}}dx[\/latex]<\/li>\r\n \t
                                                                        3. [latex]\\displaystyle\\int \\frac{\\text{arctan}\\left({x}^{3}\\right)}{{x}^{4}}dx[\/latex]<\/li>\r\n<\/ol>\r\n

                                                                          For the following exercises (21-23), use tables to perform the integration.<\/strong><\/p>\r\n\r\n

                                                                            \r\n \t
                                                                          1. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+16}}[\/latex]<\/li>\r\n \t
                                                                          2. [latex]\\displaystyle\\int \\frac{dx}{1-\\cos\\left(4x\\right)}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises (23-27), solve each problem.<\/strong>\r\n
                                                                              \r\n \t
                                                                            1. Find the area bounded by [latex]y\\left(4+25{x}^{2}\\right)=5,x=0,y=0,\\text{and }x=4[\/latex]. Use a table of integrals or a CAS.<\/li>\r\n \t
                                                                            2. Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\\le x\\le 3[\/latex], about the [latex]x[\/latex]-axis. (Round the answer to two decimal places.)<\/li>\r\n \t
                                                                            3. Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\\cos{x},0\\le x\\le \\frac{\\pi }{2}[\/latex], about the [latex]x[\/latex]-axis. (Round the answer to two decimal places.)<\/li>\r\n \t
                                                                            4. Find the length of the curve [latex]y={e}^{x}[\/latex] over [latex]\\left[0,\\text{ln}\\left(2\\right)\\right][\/latex].<\/li>\r\n \t
                                                                            5. Find the average value of the function [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}+1}[\/latex] over the interval [latex]\\left[-3,3\\right][\/latex].<\/li>\r\n<\/ol>","rendered":"

                                                                              Integration by Parts<\/h2>\n

                                                                              In the following exercises (1-3), use the guidelines in this section to choose u.<\/em> Do not <\/em>evaluate the integrals.<\/strong><\/p>\n

                                                                                \n
                                                                              1. [latex]\\displaystyle\\int {x}^{3}{e}^{2x}dx[\/latex]<\/li>\n
                                                                              2. \u00a0<\/strong>[latex]\\displaystyle\\int {y}^{3}\\cos{y} dy[\/latex]<\/li>\n
                                                                              3. [latex]\\displaystyle\\int {e}^{3x}\\sin\\left(2x\\right)dx[\/latex]<\/span><\/li>\n<\/ol>\n

                                                                                For the following exercises (4-19), find\u00a0the integral by using the simplest method. Not all problems require integration by parts.<\/em><\/strong><\/p>\n

                                                                                  \n
                                                                                1. [latex]\\displaystyle\\int \\text{ln}xdx[\/latex] (Hint:<\/em> [latex]\\displaystyle\\int \\text{ln}xdx[\/latex] is equivalent to [latex]\\displaystyle\\int 1\\cdot \\text{ln}\\left( x\\right)dx.[\/latex])<\/li>\n
                                                                                2. [latex]\\displaystyle\\int {\\tan}^{-1}xdx[\/latex]<\/li>\n
                                                                                3. \u00a0<\/strong>[latex]\\displaystyle\\int x\\sin\\left(2x\\right)dx[\/latex]<\/span><\/li>\n
                                                                                4. [latex]\\displaystyle\\int x{e}^{\\text{-}x}dx[\/latex]<\/li>\n
                                                                                5. [latex]\\displaystyle\\int {x}^{2}\\cos{x}dx[\/latex]<\/li>\n
                                                                                6. \u00a0<\/strong>[latex]\\displaystyle\\int \\text{ln}\\left(2x+1\\right)dx[\/latex]<\/li>\n
                                                                                7. [latex]\\displaystyle\\int {e}^{x}\\sin{x}dx[\/latex]<\/li>\n
                                                                                8. [latex]\\displaystyle\\int x{e}^{\\text{-}{x}^{2}}dx[\/latex]<\/li>\n
                                                                                9. \u00a0<\/strong>[latex]\\displaystyle\\int \\sin\\left(\\text{ln}\\left(2x\\right)\\right)dx[\/latex]<\/li>\n
                                                                                10. [latex]\\displaystyle\\int {\\left(\\text{ln}x\\right)}^{2}dx[\/latex]<\/li>\n
                                                                                11. \u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\text{ln}xdx[\/latex]<\/li>\n
                                                                                12. \u00a0<\/strong>[latex]\\displaystyle\\int {\\cos}^{-1}\\left(2x\\right)dx[\/latex]<\/li>\n
                                                                                13. [latex]\\displaystyle\\int {x}^{2}\\sin{x}dx[\/latex]<\/li>\n
                                                                                14. [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/li>\n
                                                                                15. [latex]\\displaystyle\\int x{\\sec}^{-1}xdx[\/latex]<\/li>\n
                                                                                16. [latex]\\displaystyle\\int x\\text{cosh}xdx[\/latex]<\/span><\/li>\n<\/ol>\n

                                                                                  For the following exercises (20-24), compute the definite integrals. Use a graphing utility to confirm your answers.<\/strong><\/p>\n

                                                                                    \n
                                                                                  1. \u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{1}x{e}^{-2x}dx[\/latex] (Express the answer in exact form.)<\/li>\n
                                                                                  2. [latex]{\\displaystyle\\int }_{1}^{e}\\text{ln}\\left({x}^{2}\\right)dx[\/latex]<\/span><\/li>\n
                                                                                  3. [latex]{\\displaystyle\\int }_{\\text{-}\\pi }^{\\pi }x\\sin{x}dx[\/latex] (Express the answer in exact form.)<\/li>\n
                                                                                  4. \u00a0<\/strong>[latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{x}^{2}\\sin{x}dx[\/latex] (Express the answer in exact form.)<\/li>\n
                                                                                  5. Evaluate [latex]\\displaystyle\\int \\cos{x}\\text{ln}\\left(\\sin{x}\\right)dx[\/latex]<\/li>\n<\/ol>\n

                                                                                    Derive the following formulas (25-26) using the technique of integration by parts. Assume that <\/em><\/span>[latex]n[\/latex] is a positive integer. These formulas are called <\/span>reduction formulas because the exponent in the <\/span>[latex]x[\/latex] term has been reduced by one in each case. The second integral is simpler than the original integral.<\/span><\/em><\/strong><\/p>\n

                                                                                      \n
                                                                                    1. [latex]\\displaystyle\\int {x}^{n}\\cos{x}dx={x}^{n}\\sin{x}-n\\displaystyle\\int {x}^{n - 1}\\sin{x}dx[\/latex]<\/li>\n
                                                                                    2. \n

                                                                                      Integrate [latex]\\displaystyle\\int 2x\\sqrt{2x - 3}dx[\/latex] using two methods:<\/p>\n

                                                                                        \n
                                                                                      1. Using parts, letting [latex]dv=\\sqrt{2x - 3}dx[\/latex]<\/li>\n
                                                                                      2. Substitution, letting [latex]u=2x - 3[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                                        In the following exercises (27-29), state whether you would use integration by parts to evaluate the integral. If so, identify [latex]u[\/latex]and [latex]dv.[\/latex] If not, describe the technique used to perform the integration without actually doing the problem.<\/strong><\/p>\n

                                                                                          \n
                                                                                        1. [latex]\\displaystyle\\int \\frac{{\\text{ln}}^{2}x}{x}dx[\/latex]<\/li>\n
                                                                                        2. \u00a0<\/strong>[latex]\\displaystyle\\int x{e}^{{x}^{2}-3}dx[\/latex]<\/span><\/li>\n
                                                                                        3. \u00a0<\/strong>[latex]\\displaystyle\\int {x}^{2}\\sin\\left(3{x}^{3}+2\\right)dx[\/latex]<\/li>\n<\/ol>\n

                                                                                          In the following exercise, sketch the region bounded above by the curve, the [latex]x[\/latex]-axis, and [latex]x=1[\/latex], and find the area of the region. Provide the exact form or round answers to the number of places indicated.<\/strong><\/p>\n

                                                                                            \n
                                                                                          1. [latex]y={e}^{\\text{-}x}\\sin\\left(\\pi x\\right)[\/latex] (Approximate answer to five decimal places.)<\/li>\n<\/ol>\n

                                                                                            Find the volume generated by rotating the region bounded by the given curves about the specified line for the following exercises (31-34). Express the answers in exact form or approximate to the number of decimal places indicated.<\/strong><\/p>\n

                                                                                              \n
                                                                                            1. [latex]y={e}^{\\text{-}x}[\/latex] [latex]y=0,x=-1x=0[\/latex]; about [latex]x=1[\/latex] (Express the answer in exact form.)<\/li>\n
                                                                                            2. Find the area under the graph of [latex]y={\\sec}^{3}x[\/latex] from [latex]x=0\\text{to}x=1[\/latex]. (Round the answer to two significant digits.)<\/li>\n
                                                                                            3. \n

                                                                                              Find the area of the region enclosed by the curve [latex]y=x\\cos{x}[\/latex] and the x<\/em>-axis for [latex]\\frac{11\\pi }{2}\\le x\\le \\frac{13\\pi }{2}[\/latex]. (Express the answer in exact form.)<\/p>\n<\/li>\n

                                                                                            4. Find the volume of the solid generated by revolving the region bounded by the curve [latex]y=4\\cos{x}[\/latex] and the x<\/em>-axis, [latex]\\frac{\\pi }{2}\\le x\\le \\frac{3\\pi }{2}[\/latex], about the x<\/em>-axis. (Express the answer in exact form.)<\/li>\n<\/ol>\n

                                                                                              Trigonometric Integrals<\/h2>\n

                                                                                              For the following exercises (1-3), evaluate each of the following integrals by u<\/em>-substitution.<\/strong><\/p>\n

                                                                                                \n
                                                                                              1. [latex]\\displaystyle\\int {\\sin}^{3}x\\cos{x}dx[\/latex]<\/li>\n
                                                                                              2. [latex]\\displaystyle\\int {\\tan}^{5}\\left(2x\\right){\\sec}^{2}\\left(2x\\right)dx[\/latex]<\/li>\n
                                                                                              3. [latex]\\displaystyle\\int \\tan\\left(\\dfrac{x}{2}\\right){\\sec}^{2}\\left(\\dfrac{x}{2}\\right)dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                In the following exercises (4-11), compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions.<\/strong><\/p>\n

                                                                                                  \n
                                                                                                1. [latex]\\displaystyle\\int {\\sin}^{3}xdx[\/latex]<\/li>\n
                                                                                                2. [latex]\\displaystyle\\int \\sin{x}\\cos{x}dx[\/latex]<\/li>\n
                                                                                                3. [latex]\\displaystyle\\int {\\sin}^{5}x{\\cos}^{2}xdx[\/latex]<\/li>\n
                                                                                                4. [latex]\\displaystyle\\int \\sqrt{\\sin{x}}\\cos{x}dx[\/latex]<\/li>\n
                                                                                                5. [latex]\\displaystyle\\int \\sec{x}\\tan{x}dx[\/latex]<\/li>\n
                                                                                                6. [latex]\\displaystyle\\int {\\tan}^{2}x\\sec{x}dx[\/latex]<\/li>\n
                                                                                                7. [latex]\\displaystyle\\int {\\sec}^{4}xdx[\/latex]<\/li>\n
                                                                                                8. [latex]\\displaystyle\\int \\csc{x}dx[\/latex]<\/li>\n<\/ol>\n

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

                                                                                                    \n
                                                                                                  1. [latex]\\displaystyle\\int {\\sin}^{2}ax\\cos{ax} dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                    Use the double-angle formulas to evaluate the following integrals (13-15).<\/strong><\/p>\n

                                                                                                      \n
                                                                                                    1. [latex]{\\displaystyle\\int }_{0}^{\\pi }{\\sin}^{2}xdx[\/latex]<\/li>\n
                                                                                                    2. [latex]\\displaystyle\\int {\\cos}^{2}3xdx[\/latex]<\/li>\n
                                                                                                    3. [latex]\\displaystyle\\int {\\sin}^{2}xdx+\\displaystyle\\int {\\cos}^{2}xdx[\/latex]<\/li>\n<\/ol>\n

                                                                                                      For the following exercises (16-22), evaluate the definite integrals. Express answers in exact form whenever possible.<\/strong><\/p>\n

                                                                                                        \n
                                                                                                      1. [latex]{\\displaystyle\\int }_{0}^{2\\pi }\\cos{x}\\sin2xdx[\/latex]<\/li>\n
                                                                                                      2. [latex]{\\displaystyle\\int }_{0}^{\\pi }\\cos\\left(99x\\right)\\sin\\left(101x\\right)dx[\/latex]<\/li>\n
                                                                                                      3. [latex]{\\displaystyle\\int }_{0}^{2\\pi }\\sin{x}\\sin\\left(2x\\right)\\sin\\left(3x\\right)dx[\/latex]<\/li>\n
                                                                                                      4. [latex]{\\displaystyle\\int }_{\\dfrac{\\pi}{6}}^{\\dfrac{\\pi}{3}}\\dfrac{{\\cos}^{3}x}{\\sqrt{\\sin{x}}}dx[\/latex] (Round this answer to three decimal places.)<\/li>\n
                                                                                                      5. [latex]{\\displaystyle\\int }_{0}^{\\dfrac{\\pi}{2}}\\sqrt{1-\\cos\\left(2x\\right)}dx[\/latex]<\/li>\n
                                                                                                      6. Find the area of the region bounded by the graphs of the equations [latex]y={\\cos}^{2}x,y={\\sin}^{2}x,x=-\\dfrac{\\pi }{4},\\text{and }x=\\dfrac{\\pi }{4}[\/latex].<\/li>\n
                                                                                                      7. 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].<\/li>\n<\/ol>\n

                                                                                                        For the following exercises (23-24), solve the differential equations.<\/strong><\/p>\n

                                                                                                          \n
                                                                                                        1. [latex]\\dfrac{dy}{d\\theta }={\\sin}^{4}\\left(\\pi \\theta \\right)[\/latex]<\/li>\n
                                                                                                        2. Find the length of the curve [latex]y=\\text{ln}\\left(\\sin{x}\\right),\\dfrac{\\pi }{3}\\le x\\le \\dfrac{\\pi }{2}[\/latex].<\/li>\n<\/ol>\n

                                                                                                          For the following exercises (25-26), use this information: <\/strong><\/p>\n

                                                                                                            \n
                                                                                                          • The inner product of two functions [latex]f[\/latex] and [latex]g[\/latex] over [latex]\\left[a,b\\right][\/latex] is defined by<\/strong><\/strong>\n
                                                                                                            [latex]f\\left(x\\right)\\cdot g\\left(x\\right)=\\langle f,g\\rangle ={\\displaystyle\\int }_{a}^{b}f\\cdot gdx[\/latex]<\/div>\n<\/li>\n
                                                                                                          • Two distinct functions [latex]f[\/latex] and [latex]g[\/latex] are said to be orthogonal if [latex]\\langle f,g\\rangle =0[\/latex].<\/strong><\/li>\n<\/ul>\n
                                                                                                              \n
                                                                                                            1. Show that [latex]{\\sin(2x),\\cos(3x)}[\/latex] are orthogonal over the interval [latex][\\text{-}\\pi ,\\pi ][\/latex].<\/li>\n
                                                                                                            2. Integrate [latex]{y}^{\\prime }=\\sqrt{\\tan{x}}{\\sec}^{4}x[\/latex].<\/li>\n<\/ol>\n

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

                                                                                                                \n
                                                                                                              1. [latex]\\displaystyle\\int {\\tan}^{350}x{\\sec}^{2}xdx[\/latex] or [latex]\\displaystyle\\int {\\tan}^{350}x\\sec{x}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                Simplify the following expressions by writing each one using a single trigonometric function.<\/strong><\/p>\n

                                                                                                                  \n
                                                                                                                1. [latex]9{\\sec}^{2}\\theta -9[\/latex]<\/li>\n
                                                                                                                2. [latex]{a}^{2}+{a}^{2}{\\text{sinh}}^{2}\\theta[\/latex]<\/li>\n<\/ol>\n

                                                                                                                  Use the technique of completing the square to express each trinomial as the square of a binomial.<\/strong><\/p>\n

                                                                                                                    \n
                                                                                                                  1. [latex]4{x}^{2}-4x+1[\/latex]<\/li>\n
                                                                                                                  2. [latex]\\text{-}{x}^{2}-2x+4[\/latex]<\/li>\n<\/ol>\n

                                                                                                                    In the following exercises (32-38), integrate using the method of trigonometric substitution. Express the final answer in terms of the variable.<\/strong><\/p>\n

                                                                                                                      \n
                                                                                                                    1. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}-{a}^{2}}}[\/latex]<\/li>\n
                                                                                                                    2. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1+9{x}^{2}}}[\/latex]<\/li>\n
                                                                                                                    3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
                                                                                                                    4. [latex]\\displaystyle\\int \\sqrt{{x}^{2}+9}dx[\/latex]<\/li>\n
                                                                                                                    5. [latex]\\displaystyle\\int \\frac{{\\theta }^{3}d\\theta }{\\sqrt{9-{\\theta }^{2}}}d\\theta[\/latex]<\/li>\n
                                                                                                                    6. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}\\sqrt{{x}^{2}+1}}[\/latex]<\/li>\n
                                                                                                                    7. [latex]{\\displaystyle\\int }_{-1}^{1}{\\left(1-{x}^{2}\\right)}^{\\frac{3}{2}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                      In the following exercises (39-41), use the substitutions [latex]x=\\text{sinh}\\theta ,\\text{cosh}\\theta[\/latex], or [latex]\\text{tanh}\\theta[\/latex]. Express the final answers in terms of the variable [latex]x[\/latex].<\/strong><\/p>\n

                                                                                                                        \n
                                                                                                                      1. [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{1-{x}^{2}}}[\/latex]<\/li>\n
                                                                                                                      2. [latex]\\displaystyle\\int \\frac{\\sqrt{{x}^{2}-1}}{{x}^{2}}dx[\/latex]<\/li>\n
                                                                                                                      3. [latex]\\displaystyle\\int \\frac{\\sqrt{1+{x}^{2}}}{{x}^{2}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                        Use the technique of completing the square to evaluate the following integrals (42-43).<\/strong><\/p>\n

                                                                                                                          \n
                                                                                                                        1. [latex]\\displaystyle\\int \\frac{1}{{x}^{2}+2x+1}dx[\/latex]<\/li>\n
                                                                                                                        2. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{\\text{-}{x}^{2}+10x}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                          For the following exercises (44-49), solve each problem.<\/strong><\/p>\n

                                                                                                                            \n
                                                                                                                          1. Evaluate the integral without using calculus: [latex]{\\displaystyle\\int }_{-3}^{3}\\sqrt{9-{x}^{2}}dx[\/latex].<\/li>\n
                                                                                                                          2. Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex] using two different substitutions. First, let [latex]x=\\cos\\theta[\/latex] and evaluate using trigonometric substitution. Second, let [latex]x=\\sin\\theta[\/latex] and use trigonometric substitution. Are the answers the same?<\/li>\n
                                                                                                                          3. Evaluate the integral [latex]\\displaystyle\\int \\frac{x}{{x}^{2}+1}dx[\/latex] using the form [latex]\\displaystyle\\int \\frac{1}{u}du[\/latex]. Next, evaluate the same integral using [latex]x=\\tan\\theta[\/latex]. Are the results the same?<\/li>\n
                                                                                                                          4. State the method of integration you would use to evaluate the integral [latex]\\displaystyle\\int {x}^{2}\\sqrt{{x}^{2}-1}dx[\/latex]. Why did you choose this method?<\/li>\n
                                                                                                                          5. Find the length of the arc of the curve over the specified interval: [latex]y=\\text{ln}x,\\left[1,5\\right][\/latex]. Round the answer to three decimal places.<\/li>\n
                                                                                                                          6. The region bounded by the graph of [latex]f\\left(x\\right)=\\frac{1}{1+{x}^{2}}[\/latex] and the [latex]x[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex] is revolved about the [latex]x[\/latex]-axis. Find the volume of the solid that is generated.<\/li>\n<\/ol>\n

                                                                                                                            For the following exercises (50-52), solve the initial-value problem for [latex]y[\/latex] as a function of [latex]x[\/latex].<\/strong><\/p>\n

                                                                                                                              \n
                                                                                                                            1. [latex]\\left(64-{x}^{2}\\right)\\frac{dy}{dx}=1,y\\left(0\\right)=3[\/latex]<\/li>\n
                                                                                                                            2. An oil storage tank can be described as the volume generated by revolving the area bounded by [latex]y=\\frac{16}{\\sqrt{64+{x}^{2}}},x=0,y=0,x=2[\/latex] about the [latex]x[\/latex]-axis. Find the volume of the tank (in cubic meters).<\/li>\n
                                                                                                                            3. Find the length of the curve [latex]y=\\sqrt{16-{x}^{2}}[\/latex] between [latex]x=0[\/latex] and [latex]x=2[\/latex].<\/li>\n<\/ol>\n

                                                                                                                              Partial Fractions<\/h2>\n

                                                                                                                              In the following exercises (1-7), express the rational function as a sum or difference of two simpler rational expressions.<\/strong><\/p>\n

                                                                                                                                \n
                                                                                                                              1. [latex]\\frac{{x}^{2}+1}{x\\left(x+1\\right)\\left(x+2\\right)}[\/latex]<\/li>\n
                                                                                                                              2. [latex]\\frac{3x+1}{{x}^{2}}[\/latex]<\/li>\n
                                                                                                                              3. [latex]\\frac{2{x}^{4}}{{x}^{2}-2x}[\/latex]<\/li>\n
                                                                                                                              4. [latex]\\frac{1}{{x}^{2}\\left(x - 1\\right)}[\/latex]<\/li>\n
                                                                                                                              5. [latex]\\frac{1}{x\\left(x - 1\\right)\\left(x - 2\\right)\\left(x - 3\\right)}[\/latex]<\/li>\n
                                                                                                                              6. [latex]\\frac{3{x}^{2}}{{x}^{3}-1}=\\frac{3{x}^{2}}{\\left(x - 1\\right)\\left({x}^{2}+x+1\\right)}[\/latex]<\/li>\n
                                                                                                                              7. [latex]\\frac{3{x}^{4}+{x}^{3}+20{x}^{2}+3x+31}{\\left(x+1\\right){\\left({x}^{2}+4\\right)}^{2}}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                Use the method of partial fractions to evaluate each of the following integrals (8-12).<\/strong><\/p>\n

                                                                                                                                  \n
                                                                                                                                1. [latex]\\displaystyle\\int \\frac{3x}{{x}^{2}+2x - 8}dx[\/latex]<\/li>\n
                                                                                                                                2. [latex]\\displaystyle\\int \\frac{x}{{x}^{2}-4}dx[\/latex]<\/li>\n
                                                                                                                                3. [latex]\\displaystyle\\int \\frac{2{x}^{2}+4x+22}{{x}^{2}+2x+10}dx[\/latex]<\/li>\n
                                                                                                                                4. [latex]\\displaystyle\\int \\frac{2-x}{{x}^{2}+x}dx[\/latex]<\/li>\n
                                                                                                                                5. [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}-2{x}^{2}-4x+8}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                  Evaluate the following integrals (13-14), which have irreducible quadratic factors.<\/strong><\/p>\n

                                                                                                                                    \n
                                                                                                                                  1. [latex]\\displaystyle\\int \\frac{2}{\\left(x - 4\\right)\\left({x}^{2}+2x+6\\right)}dx[\/latex]<\/li>\n
                                                                                                                                  2. [latex]\\displaystyle\\int \\frac{{x}^{3}+6{x}^{2}+3x+6}{{x}^{3}+2{x}^{2}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                    Use the method of partial fractions to evaluate the following integrals (15-16).<\/strong><\/p>\n

                                                                                                                                      \n
                                                                                                                                    1. [latex]\\displaystyle\\int \\frac{3x+4}{\\left({x}^{2}+4\\right)\\left(3-x\\right)}dx[\/latex]<\/li>\n
                                                                                                                                    2. [latex]\\displaystyle\\int \\frac{3x+4}{{x}^{3}-2x - 4}dx[\/latex] (Hint: Use the rational root theorem.)<\/li>\n<\/ol>\n

                                                                                                                                      Use substitution to convert the integrals to integrals of rational functions. Then use partial fractions to evaluate the integrals.<\/strong><\/p>\n

                                                                                                                                        \n
                                                                                                                                      1. [latex]\\displaystyle\\int \\frac{{e}^{x}dx}{{e}^{2x}-{e}^{x}}dx[\/latex]<\/li>\n
                                                                                                                                      2. [latex]\\displaystyle\\int \\frac{\\sin{x}}{{\\cos}^{2}x+\\cos{x} - 6}dx[\/latex]<\/li>\n
                                                                                                                                      3. [latex]\\displaystyle\\int \\frac{dt}{{\\left({e}^{t}-{e}^{\\text{-}t}\\right)}^{2}}[\/latex]<\/li>\n
                                                                                                                                      4. [latex]\\displaystyle\\int \\frac{dx}{1+\\sqrt{x+1}}[\/latex]<\/li>\n
                                                                                                                                      5. [latex]\\displaystyle\\int \\frac{\\cos{x}}{\\sin{x}\\left(1-\\sin{x}\\right)}dx[\/latex]<\/li>\n
                                                                                                                                      6. [latex]\\underset{1}{\\overset{2}{\\displaystyle\\int }}\\frac{1}{{x}^{2}\\sqrt{4-{x}^{2}}}dx[\/latex]<\/li>\n
                                                                                                                                      7. [latex]\\displaystyle\\int \\frac{1}{1+{e}^{x}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                        For the following exercises (44-49), use the given substitution to convert the integral to an integral of a rational function, then evaluate.<\/strong><\/p>\n

                                                                                                                                          \n
                                                                                                                                        1. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{x}+\\sqrt[3]{x}}dx;x={u}^{6}[\/latex]<\/li>\n
                                                                                                                                        2. Find the volume of the solid generated when the region bounded by [latex]y=\\frac{1}{\\sqrt{x\\left(3-x\\right)}}[\/latex], [latex]y=0[\/latex], [latex]x=1[\/latex], and [latex]x=2[\/latex] is revolved about the x-axis.<\/li>\n<\/ol>\n

                                                                                                                                          In the following exercises (1-7), solve the initial-value problem for [latex]x[\/latex] as a function of [latex]t[\/latex].<\/strong><\/p>\n

                                                                                                                                            \n
                                                                                                                                          1. [latex]\\left({t}^{2}-7t+12\\right)\\frac{dx}{dt}=1,\\left(t>4,x\\left(5\\right)=0\\right)[\/latex]<\/li>\n
                                                                                                                                          2. [latex]\\left(2{t}^{3}-2{t}^{2}+t - 1\\right)\\frac{dx}{dt}=3,x\\left(2\\right)=0[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                            For the following exercises (28-31), solve each problem.<\/strong><\/p>\n

                                                                                                                                              \n
                                                                                                                                            1. Find the volume generated by revolving the area bounded by [latex]y=\\frac{1}{{x}^{3}+7{x}^{2}+6x}x=1,x=7,\\text{and }y=0[\/latex] about the y-axis.<\/li>\n
                                                                                                                                            2. Evaluate the integral [latex]\\displaystyle\\int \\frac{dx}{{x}^{3}+1}[\/latex].<\/li>\n
                                                                                                                                            3. Find the area under the curve [latex]y=\\frac{1}{1+\\sin{x}}[\/latex] between [latex]x=0[\/latex] and [latex]x=\\pi[\/latex]. (Assume the dimensions are in inches.)<\/li>\n
                                                                                                                                            4. Evaluate [latex]\\displaystyle\\int \\frac{\\sqrt[3]{x - 8}}{x}dx[\/latex].<\/li>\n<\/ol>\n

                                                                                                                                              Other Strategies for Integration<\/h2>\n

                                                                                                                                              For the following exercises (1-8), use a table of integrals to evaluate the following integrals.<\/strong><\/p>\n

                                                                                                                                                \n
                                                                                                                                              1. [latex]\\displaystyle\\int \\frac{x+3}{{x}^{2}+2x+2}dx[\/latex]<\/li>\n
                                                                                                                                              2. [latex]\\displaystyle\\int \\frac{1}{\\sqrt{{x}^{2}+6x}}dx[\/latex]<\/li>\n
                                                                                                                                              3. [latex]\\displaystyle\\int x\\cdot {2}^{{x}^{2}}dx[\/latex]<\/li>\n
                                                                                                                                              4. [latex]\\displaystyle\\int \\frac{dy}{\\sqrt{4-{y}^{2}}}[\/latex]<\/li>\n
                                                                                                                                              5. [latex]\\displaystyle\\int \\csc\\left(2w\\right)\\cot\\left(2w\\right)dw[\/latex]<\/li>\n
                                                                                                                                              6. [latex]{\\displaystyle\\int }_{0}^{1}\\frac{3xdx}{\\sqrt{{x}^{2}+8}}[\/latex]<\/li>\n
                                                                                                                                              7. [latex]{\\displaystyle\\int }_{0}^{\\frac{\\pi}{2}}{\\tan}^{2}\\left(\\frac{x}{2}\\right)dx[\/latex]<\/li>\n
                                                                                                                                              8. [latex]\\displaystyle\\int {\\tan}^{5}\\left(3x\\right)dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                Use a CAS to evaluate the following integrals (9-14). Tables can also be used to verify the answers.<\/strong><\/p>\n

                                                                                                                                                  \n
                                                                                                                                                1. [latex]\\displaystyle\\int \\frac{dw}{1+\\sec\\left(\\frac{w}{2}\\right)}[\/latex]<\/li>\n
                                                                                                                                                2. [latex]{\\displaystyle\\int }_{0}^{t}\\frac{dt}{4\\cos{t}+3\\sin{t}}[\/latex]<\/li>\n
                                                                                                                                                3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{\\frac{1}{2}}+{x}^{\\frac{1}{3}}}[\/latex]<\/li>\n
                                                                                                                                                4. [latex]\\displaystyle\\int {x}^{3}\\sin{x}dx[\/latex]<\/li>\n
                                                                                                                                                5. [latex]\\displaystyle\\int \\frac{x}{1+{e}^{\\text{-}{x}^{2}}}dx[\/latex]<\/li>\n
                                                                                                                                                6. [latex]\\displaystyle\\int \\frac{dx}{x\\sqrt{x - 1}}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                  Use a calculator or CAS to evaluate the following integrals (15-17).<\/strong><\/p>\n

                                                                                                                                                    \n
                                                                                                                                                  1. [latex]{\\displaystyle\\int }_{0}^{\\frac{z\\pi}{4}}\\cos\\left(2x\\right)dx[\/latex]<\/li>\n
                                                                                                                                                  2. [latex]{\\displaystyle\\int }_{0}^{8}\\frac{2x}{\\sqrt{{x}^{2}+36}}dx[\/latex]<\/li>\n
                                                                                                                                                  3. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+4x+13}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                    Use tables to evaluate the following integrals (18-20). You may need to complete the square or change variables to put the integral into a form given in the table.<\/strong><\/p>\n

                                                                                                                                                      \n
                                                                                                                                                    1. [latex]\\displaystyle\\int \\frac{dx}{{x}^{2}+2x+10}[\/latex]<\/li>\n
                                                                                                                                                    2. [latex]\\displaystyle\\int \\frac{{e}^{x}}{\\sqrt{{e}^{2x}-4}}dx[\/latex]<\/li>\n
                                                                                                                                                    3. [latex]\\displaystyle\\int \\frac{\\text{arctan}\\left({x}^{3}\\right)}{{x}^{4}}dx[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                      For the following exercises (21-23), use tables to perform the integration.<\/strong><\/p>\n

                                                                                                                                                        \n
                                                                                                                                                      1. [latex]\\displaystyle\\int \\frac{dx}{\\sqrt{{x}^{2}+16}}[\/latex]<\/li>\n
                                                                                                                                                      2. [latex]\\displaystyle\\int \\frac{dx}{1-\\cos\\left(4x\\right)}[\/latex]<\/li>\n<\/ol>\n

                                                                                                                                                        For the following exercises (23-27), solve each problem.<\/strong><\/p>\n

                                                                                                                                                          \n
                                                                                                                                                        1. Find the area bounded by [latex]y\\left(4+25{x}^{2}\\right)=5,x=0,y=0,\\text{and }x=4[\/latex]. Use a table of integrals or a CAS.<\/li>\n
                                                                                                                                                        2. Use substitution and a table of integrals to find the area of the surface generated by revolving the curve [latex]y={e}^{x},0\\le x\\le 3[\/latex], about the [latex]x[\/latex]-axis. (Round the answer to two decimal places.)<\/li>\n
                                                                                                                                                        3. Use a CAS or tables to find the area of the surface generated by revolving the curve [latex]y=\\cos{x},0\\le x\\le \\frac{\\pi }{2}[\/latex], about the [latex]x[\/latex]-axis. (Round the answer to two decimal places.)<\/li>\n
                                                                                                                                                        4. Find the length of the curve [latex]y={e}^{x}[\/latex] over [latex]\\left[0,\\text{ln}\\left(2\\right)\\right][\/latex].<\/li>\n
                                                                                                                                                        5. Find the average value of the function [latex]f\\left(x\\right)=\\frac{1}{{x}^{2}+1}[\/latex] over the interval [latex]\\left[-3,3\\right][\/latex].<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":30,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":541,"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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/737"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions"}],"predecessor-version":[{"id":1739,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/737\/revisions\/1739"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/541"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/737\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=737"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=737"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}