1. 1
   Let $f(x) = x\,e^{3x}$, for $x \in \mathbb{R}$.

   (a)

   Find $\displaystyle\int f(x)\,dx$.

   [1 mark]

   • A$\left(\dfrac{x}{3} + \dfrac{1}{9}\right)e^{3x} + C$
   • B$\left(\dfrac{x}{3} - \dfrac{1}{3}\right)e^{3x} + C$
   • C$\dfrac{x^{2}}{6}e^{3x} + C$
   • D$\left(\dfrac{x}{3} - \dfrac{1}{9}\right)e^{3x} + C$

   (b)

   Find the exact value of $\displaystyle\int_{0}^{1} f(x)\,dx$.

   [1 mark]

   • A$\dfrac{2e^{3} - 1}{9}$
   • B$\dfrac{2e^{3}}{9}$
   • C$\dfrac{2e^{3} + 1}{9}$
   • D$\dfrac{1}{3}$

   (c)

   Find the exact area of the region enclosed by the curve $y = f(x)$, the $x$-axis and the line $x = -1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $I = \displaystyle\int_{1}^{6} x\sqrt{x + 3}\,dx$. The substitution $u = x + 3$ is to be used.

   (a)

   Which integral is equal to $I$?

   [1 mark]

   • A$\displaystyle\int_{1}^{6}(u - 3)\sqrt{u}\,du$
   • B$\displaystyle\int_{4}^{9}(u - 3)\sqrt{u}\,du$
   • C$\displaystyle\int_{4}^{9}(u + 3)\sqrt{u}\,du$
   • D$\displaystyle\int_{4}^{9}u\sqrt{u}\,du$

   (b)

   Find the value of $I$.

   [1 mark]

   • A$\dfrac{232}{5}$
   • B$\dfrac{422}{5}$
   • C$\dfrac{612}{5}$
   • D$\dfrac{8 + 12\sqrt{6}}{5}$

   (c)

   Find $\displaystyle\int x\sqrt{x + 3}\,dx$, giving your answer in terms of $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $g(x) = \ln x$, for $x > 0$.

   (a)

   Use integration by parts to show that $\displaystyle\int g(x)\,dx = x\ln x - x + C$.

   [3 marks]

   (b)

   The region $R$ is enclosed by the curve $y = g(x)$, the $x$-axis and the line $x = e^{2}$. Find the exact area of $R$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a straight line. Its velocity, $v\text{ m s}^{-1}$, at time $t$ seconds is given by $v(t) = 10e^{-t}\sin t$, for $t \ge 0$.

   (a)

   Use integration by parts twice to show that $\displaystyle\int e^{-t}\sin t\,dt = -\dfrac{1}{2}e^{-t}(\sin t + \cos t) + C$.

   [6 marks]

   (b)

   Find the total distance travelled by the particle in the first $2\pi$ seconds, showing that it is $5\left(1 + e^{-\pi}\right)^{2}$ metres.

   [6 marks]

   Total for question 4: 12 marks

End of questions