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Q1
Answer
Find $\displaystyle\int \sinh^2 x \, \mathrm{d}x$.
    $\dfrac{1}{4}\sinh 2x - \dfrac{1}{2}x + C$
    Q2
    Answer
    Find $\displaystyle\int x\cosh x \, \mathrm{d}x$.
      $x\sinh x - \cosh x + C$
      Q3
      Answer
      Find $\displaystyle\int \dfrac{1}{\sqrt{x^2 + 9}} \, \mathrm{d}x$.
        $\mathrm{arsinh}\left(\dfrac{x}{3}\right) + C = \ln\left(x + \sqrt{x^2 + 9}\right) + C$
        Q4
        Answer
        Find $\displaystyle\int \dfrac{1}{\sqrt{x^2 + 2x - 3}} \, \mathrm{d}x$, for $x > 2$.
          $\mathrm{arcosh}\left(\dfrac{x+1}{2}\right) + C = \ln\left(x + 1 + \sqrt{(x+1)^2 - 4}\right) + C$
          Q5
          Answer
          Find $\displaystyle\int \dfrac{1}{\sqrt{-x^2 + 4x + 12}} \, \mathrm{d}x$.
            $\arcsin\left(\dfrac{x-2}{4}\right) + C$
            Q6
            Answer
            Find $\displaystyle\int \dfrac{1}{3x^2 + 6x + 9} \, \mathrm{d}x$.
              $\dfrac{1}{3\sqrt{2}}\arctan\left(\dfrac{x+1}{\sqrt{2}}\right) + C$
              Q7
              Answer
              Evaluate $\displaystyle\int_0^1 2\,\mathrm{arsinh}\, 3x \, \mathrm{d}x$, giving your answer in exact form.
                $2\ln(3 + \sqrt{10}) - \dfrac{2\sqrt{10}}{3} + \dfrac{2}{3}$
                Q8
                Answer
                The region $R$ is bounded by the curve $y = \cosh x$, the $x$-axis, and the lines $x = 0$ and $x = \ln 3$. Find the exact volume of the solid of revolution formed when $R$ is rotated through $2\pi$ radians about the $x$-axis.
                  $\pi\left(\dfrac{\ln 3}{2} + \dfrac{20}{9}\right)$
                  Q9
                  Answer
                  In this question you must show detailed reasoning.
                  1. Explain why $\displaystyle\int_0^{\infty} \dfrac{1}{x^2 + 4} \, \mathrm{d}x$ is an improper integral.
                    1. Evaluate the integral exactly.
                      1. The upper limit of integration is $\infty$
                      2. $\dfrac{\pi}{4}$
                      Q10
                      Answer
                      In this question you must show detailed reasoning.
                      1. Explain why $\displaystyle\int_0^3 \dfrac{1}{(x-1)^{\frac{2}{3}}} \, \mathrm{d}x$ is an improper integral.
                        1. Evaluate the integral exactly
                          1. The integrand is undefined at $x = 1$, which lies within the interval
                          2. $3 + 3\sqrt[3]{2}$
                          Q11
                          Answer
                          In this question you must show detailed reasoning.
                          1. Explain why $\displaystyle\int_0^{\infty} \dfrac{1}{\sqrt{4x^2 + 9}} \, \mathrm{d}x$ is an improper integral.
                            1. Evaluate the integral .
                              1. The upper limit of integration is $\infty$, so the interval of integration is unbounded.
                              2. Diverges
                              Q12
                              Answer
                              In this question you must show detailed reasoning.
                              1. Explain why $\displaystyle\int_1^{\infty} \dfrac{\arctan x}{1 + x^2} \, \mathrm{d}x$ is an improper integral.
                                1. Evaluate the integral exactly.
                                  1. Check the limits
                                  2. $\dfrac{3\pi^2}{32}$