{"id":419,"date":"2025-02-13T19:44:45","date_gmt":"2025-02-13T19:44:45","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/techniques-for-integration-get-stronger\/"},"modified":"2025-02-13T19:44:45","modified_gmt":"2025-02-13T19:44:45","slug":"techniques-for-integration-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/techniques-for-integration-get-stronger\/","title":{"raw":"Techniques for Integration: Get Stronger","rendered":"Techniques for Integration: Get Stronger"},"content":{"raw":"\n

Substitution<\/h2>\n
    \n\t
  1. Why is [latex]u[\/latex]-substitution referred to as change of variable<\/em>?<\/li>\n\t
  2. If [latex]f=g\\circ h,[\/latex] when reversing the chain rule, [latex]\\frac{d}{dx}(g\\circ h)(x)={g}^{\\prime }(h(x)){h}^{\\prime }(x),[\/latex] should you take [latex]u=g(x)[\/latex] or [latex]u=h(x)?[\/latex]<\/li>\n<\/ol>\n

    In the following exercises, verify each identity using differentiation. Then, using the indicated [latex]u[\/latex]-substitution, identify [latex]f[\/latex] such that the integral takes the form [latex]\\displaystyle\\int f(u)du.[\/latex]<\/strong><\/p>\n

      \n\t
    1. For [latex]x>1[\/latex]:  [latex]\\displaystyle\\int \\frac{{x}^{2}}{\\sqrt{x-1}}dx = \\frac{2}{15}\\sqrt{x-1}(3{x}^{2}+4x+8)+C; \\,\\,\\, u=x-1[\/latex]<\/li>\n\t
    2. [latex]\\displaystyle\\int \\frac{x}{\\sqrt{4{x}^{2}+9}}dx=\\frac{1}{4}\\sqrt{4{x}^{2}+9}+C;\\,\\,\\, u=4{x}^{2}+9[\/latex]<\/li>\n\t
    3. [latex]\\displaystyle\\int {(x+1)}^{4}dx; \\,\\,\\, u=x+1[\/latex]<\/li>\n\t
    4. [latex]\\displaystyle\\int {(2x-3)}^{-7}dx; \\,\\,\\, u=2x-3[\/latex]<\/li>\n\t
    5. [latex]\\displaystyle\\int \\frac{x}{\\sqrt{{x}^{2}+1}}dx; \\,\\,\\, u={x}^{2}+1[\/latex]<\/li>\n\t
    6. [latex]\\displaystyle\\int (x-1){({x}^{2}-2x)}^{3}dx; \\,\\,\\, u={x}^{2}-2x[\/latex]<\/li>\n\t
    7. [latex]\\displaystyle\\int { \\cos }^{3}\\theta d\\theta ; \\,\\,\\, u= \\sin \\theta [\/latex]\n\n\n

      [reveal-answer q=\"41198867\"]Hint[\/reveal-answer]
      \n[hidden-answer a=\"41198867\"]<\/p>\n

      ([latex]\\cos ^{2}\\theta =1-{ \\sin }^{2}\\theta [\/latex])<\/p>\n

      [\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n

      In the following exercises, use a suitable change of variables to determine the indefinite integral.<\/strong><\/p>\n

        \n\t
      1. [latex]\\displaystyle\\int x{(1-x)}^{99}dx[\/latex]<\/li>\n\t
      2. [latex]\\displaystyle\\int {(11x-7)}^{-3}dx[\/latex]<\/li>\n\t
      3. [latex]\\displaystyle\\int { \\cos }^{3}\\theta \\sin \\theta d\\theta [\/latex]<\/li>\n\t
      4. [latex]\\displaystyle\\int { \\cos }^{2}(\\pi t) \\sin (\\pi t)dt[\/latex]<\/li>\n\t
      5. [latex]\\displaystyle\\int t \\sin ({t}^{2}) \\cos ({t}^{2})dt[\/latex]<\/li>\n\t
      6. [latex]\\displaystyle\\int \\frac{{x}^{2}}{{({x}^{3}-3)}^{2}}dx[\/latex]<\/li>\n\t
      7. [latex]\\displaystyle\\int \\frac{{y}^{5}}{{(1-{y}^{3})}^{3\\text{\/}2}}dy[\/latex]<\/li>\n\t
      8. [latex]{\\displaystyle\\int (1-{ \\cos }^{3}\\theta )}^{10}{ \\cos }^{2}\\theta \\sin \\theta d\\theta [\/latex]<\/li>\n\t
      9. [latex]\\displaystyle\\int ({ \\sin }^{2}\\theta -2 \\sin \\theta ){({ \\sin }^{3}\\theta -3{ \\sin }^{2}\\theta )}^{3} \\cos \\theta d\\theta [\/latex]<\/li>\n<\/ol>\n

        In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer.<\/strong><\/p>\n

          \n\t
        1. [latex]y=x{(1-{x}^{2})}^{3}[\/latex] over [latex]\\left[-1,2\\right][\/latex]<\/li>\n\t
        2. [latex]y=\\dfrac{x}{{({x}^{2}+1)}^{2}}[\/latex] over [latex]\\left[-1,1\\right][\/latex]<\/li>\n<\/ol>\n

          In the following exercises, use a change of variables to evaluate the definite integral.<\/strong><\/p>\n

            \n\t
          1. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{x}{\\sqrt{1+{x}^{2}}}dx[\/latex]<\/li>\n\t
          2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{t^2}{\\sqrt{1+{t}^{3}}}dt[\/latex]<\/li>\n\t
          3. [latex]{\\displaystyle\\int }_{0}^{\\pi \\text{\/}4}\\dfrac{ \\sin \\theta }{{ \\cos }^{4}\\theta }d\\theta [\/latex]<\/li>\n<\/ol>\n

            In the following exercises, evaluate the indefinite integral [latex]\\displaystyle\\int f(x)dx[\/latex] with constant [latex]C=0[\/latex] using [latex]u[\/latex]-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of C<\/em> that would need to be added to the antiderivative to make it equal to the definite integral [latex]F(x)={\\displaystyle\\int }_{a}^{x}f(t)dt,[\/latex] with [latex]a[\/latex] the left endpoint of the given interval.<\/strong><\/p>\n

              \n\t
            1. [latex]\\displaystyle\\int \\frac{ \\cos (\\text{ln}(2x))}{x}dx[\/latex] on [latex]\\left[0,2\\right][\/latex]<\/li>\n\t
            2. [latex]\\displaystyle\\int \\frac{ \\sin x}{{ \\cos }^{3}x}dx[\/latex] over [latex]\\left[-\\frac{\\pi }{3},\\frac{\\pi }{3}\\right][\/latex]<\/li>\n\t
            3. [latex]\\displaystyle\\int 3{x}^{2}\\sqrt{2{x}^{3}+1}dx[\/latex] over [latex]\\left[0,1\\right][\/latex]<\/li>\n\t
            4. Is the substitution [latex]u=1-{x}^{2}[\/latex] in the definite integral [latex]{\\displaystyle\\int }_{0}^{2}\\dfrac{x}{1-{x}^{2}}dx[\/latex] okay? If not, why not?<\/li>\n<\/ol>\n

              In the following exercises, use a change of variables to show that each definite integral is equal to zero.<\/strong><\/p>\n

                \n\t
              1. [latex]{\\displaystyle\\int }_{0}^{\\sqrt{\\pi }}t \\cos ({t}^{2}) \\sin ({t}^{2})dt[\/latex]<\/li>\n\t
              2. [latex]{\\displaystyle\\int }_{0}^{1}\\dfrac{1-2t}{(1+{(t-\\frac{1}{2})}^{2})}dt[\/latex]<\/li>\n\t
              3. [latex]{\\displaystyle\\int }_{0}^{2}(1-t) \\cos (\\pi t)dt[\/latex]<\/li>\n\t
              4. Show that the average value of [latex]f(x)[\/latex] over an interval [latex]\\left[a,b\\right][\/latex] is the same as the average value of [latex]f(cx)[\/latex] over the interval [latex]\\left[\\frac{a}{c},\\frac{b}{c}\\right][\/latex] for [latex]c>0.[\/latex]<\/li>\n\t
              5. Find the area under the graph of [latex]g(t)=\\dfrac{t}{{(1-{t}^{2})}^{a}}[\/latex] between [latex]t=0[\/latex] and [latex]t=x,[\/latex] where [latex]0<x<1[\/latex] and [latex]a>0[\/latex] is fixed. Evaluate the limit as [latex]x\\to 1.[\/latex]<\/li>\n\t
              6. The area of the top half of an ellipse with a major axis that is the [latex]x[\/latex]-axis from [latex]x=-1[\/latex] to [latex]a[\/latex] and with a minor axis that is the [latex]y[\/latex]-axis from [latex]y=\\text{\u2212}b[\/latex] to [latex]b[\/latex] can be written as [latex]{\\displaystyle\\int }_{\\text{\u2212}a}^{a}b\\sqrt{1-\\frac{{x}^{2}}{{a}^{2}}}dx.[\/latex] Use the substitution [latex]x=a \\cos t[\/latex] to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.<\/li>\n\t
              7. The following graph is of a function of the form [latex]f(x)=a \\cos (nt)+b \\cos (mt).[\/latex] Estimate the coefficients [latex]a[\/latex] and [latex]b[\/latex] and the frequency parameters [latex]n[\/latex] and [latex]m[\/latex]. Use these estimates to approximate [latex]{\\displaystyle\\int }_{0}^{\\pi }f(t)dt.[\/latex]\n\n\n

                \"The<\/span><\/p>\n<\/li>\n<\/ol>\n

                Integrals Involving Exponential and Logarithmic Functions<\/h2>\n

                In the following exercises, verify by differentiation that [latex]\\displaystyle\\int \\text{ln}xdx=x(\\text{ln}x-1)+C,[\/latex] then use appropriate changes of variables to compute the integral.<\/strong><\/p>\n

                  \n\t
                1. [latex]\\displaystyle\\int {x}^{2}{\\text{ln}}^{2}xdx[\/latex] <\/li>\n\t
                2. [latex]\\displaystyle\\int \\frac{\\text{ln}x}{\\sqrt{x}}dx[\/latex][latex](\\text{Hint: }\\text{ Set }u=\\sqrt{x}\\text{.})[\/latex] <\/li>\n\t
                3. Write an integral to express the area under the graph of [latex]y={e}^{t}[\/latex] between [latex]t=0[\/latex] and [latex]t=\\text{ln}x,[\/latex] and evaluate the integral.<\/li>\n<\/ol>\n

                  In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms.<\/strong><\/p>\n

                    \n\t
                  1. [latex]\\displaystyle\\int \\frac{ \\sin (3x)- \\cos (3x)}{ \\sin (3x)+ \\cos (3x)}dx[\/latex]<\/li>\n\t
                  2. [latex]\\displaystyle\\int x \\csc ({x}^{2})dx[\/latex] <\/li>\n\t
                  3. [latex]\\displaystyle\\int \\text{ln}( \\csc x) \\cot xdx[\/latex]<\/li>\n<\/ol>\n

                    In the following exercise, evaluate the definite integral.<\/strong><\/p>\n

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

                      In the following exercises, integrate using the indicated substitution.<\/strong><\/p>\n

                        \n\t
                      1. [latex]\\displaystyle\\int \\frac{y-1}{y+1}dy;u=y+1[\/latex] <\/li>\n\t
                      2. [latex]\\displaystyle\\int \\frac{ \\sin x+ \\cos x}{ \\sin x- \\cos x}dx;u= \\sin x- \\cos x[\/latex] <\/li>\n\t
                      3. [latex]\\displaystyle\\int \\text{ln}(x)\\frac{\\sqrt{1-{(\\text{ln}x)}^{2}}}{x}dx;u=\\text{ln}x[\/latex] <\/li>\n<\/ol>\n

                        In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate R<\/em>50<\/sub> and solve for the exact area.<\/strong><\/p>\n

                          \n\t
                        1. [latex]y={e}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex] <\/li>\n\t
                        2. [latex]y=\\frac{x+1}{{x}^{2}+2x+6}[\/latex] over [latex]\\left[0,1\\right][\/latex] <\/li>\n\t
                        3. [latex]y=\\text{\u2212}{2}^{\\text{\u2212}x}[\/latex] over [latex]\\left[0,1\\right][\/latex] <\/li>\n<\/ol>\n

                          Integrals Resulting in Inverse Trigonometric<\/h2>\n

                          In the following exercises, evaluate each integral in terms of an inverse trigonometric function.<\/p>\n

                            \n\t
                          1. \n[latex]{\\displaystyle\\int }_{0}^{\\sqrt{3}\\text{\/}2}\\frac{dx}{\\sqrt{1-{x}^{2}}}[\/latex]\n\n<\/li>\n\t
                          2. [latex]{\\displaystyle\\int }_{\\sqrt{3}}^{1}\\frac{dx}{\\sqrt{1+{x}^{2}}}[\/latex]\n<\/li>\n\t
                          3. [latex]{\\displaystyle\\int }_{1}^{\\sqrt{2}}\\frac{dx}{|x|\\sqrt{{x}^{2}-1}}[\/latex]\n\n<\/li>\n<\/ol>\n\n","rendered":"

                            Substitution<\/h2>\n
                              \n
                            1. Why is [latex]u[\/latex]-substitution referred to as change of variable<\/em>?<\/li>\n
                            2. If [latex]f=g\\circ h,[\/latex] when reversing the chain rule, [latex]\\frac{d}{dx}(g\\circ h)(x)={g}^{\\prime }(h(x)){h}^{\\prime }(x),[\/latex] should you take [latex]u=g(x)[\/latex] or [latex]u=h(x)?[\/latex]<\/li>\n<\/ol>\n

                              In the following exercises, verify each identity using differentiation. Then, using the indicated [latex]u[\/latex]-substitution, identify [latex]f[\/latex] such that the integral takes the form [latex]\\displaystyle\\int f(u)du.[\/latex]<\/strong><\/p>\n

                                \n
                              1. For [latex]x>1[\/latex]:  [latex]\\displaystyle\\int \\frac{{x}^{2}}{\\sqrt{x-1}}dx = \\frac{2}{15}\\sqrt{x-1}(3{x}^{2}+4x+8)+C; \\,\\,\\, u=x-1[\/latex]<\/li>\n
                              2. [latex]\\displaystyle\\int \\frac{x}{\\sqrt{4{x}^{2}+9}}dx=\\frac{1}{4}\\sqrt{4{x}^{2}+9}+C;\\,\\,\\, u=4{x}^{2}+9[\/latex]<\/li>\n
                              3. [latex]\\displaystyle\\int {(x+1)}^{4}dx; \\,\\,\\, u=x+1[\/latex]<\/li>\n
                              4. [latex]\\displaystyle\\int {(2x-3)}^{-7}dx; \\,\\,\\, u=2x-3[\/latex]<\/li>\n
                              5. [latex]\\displaystyle\\int \\frac{x}{\\sqrt{{x}^{2}+1}}dx; \\,\\,\\, u={x}^{2}+1[\/latex]<\/li>\n
                              6. [latex]\\displaystyle\\int (x-1){({x}^{2}-2x)}^{3}dx; \\,\\,\\, u={x}^{2}-2x[\/latex]<\/li>\n
                              7. [latex]\\displaystyle\\int { \\cos }^{3}\\theta d\\theta ; \\,\\,\\, u= \\sin \\theta[\/latex]\n