1. 1
   In this question $x>0$, and $C$ denotes an arbitrary constant of integration.

   (a)

   Find $\displaystyle\int\left(3\sqrt{x}+\frac{2}{x}\right)dx$.

   [1 mark]

   • A$\dfrac{3}{2\sqrt{x}}-\dfrac{2}{x^{2}}+C$
   • B$2x^{3/2}+2\ln x+C$
   • C$\dfrac{9}{2}x^{3/2}+2\ln x+C$
   • D$\dfrac{2}{3}x^{3/2}+2\ln x+C$

   (b)

   Find $\displaystyle\int\cos(2x+3)\,dx$.

   [1 mark]

   • A$\dfrac{1}{2}\sin(2x+3)+C$
   • B$2\sin(2x+3)+C$
   • C$-\dfrac{1}{2}\sin(2x+3)+C$
   • D$\sin(2x+3)+C$

   (c)

   Find $\displaystyle\int\left(e^{1-2x}+\frac{4}{2x+1}\right)dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Integration by inspection (the reverse chain rule) or by substitution may be used throughout this question. $C$ denotes an arbitrary constant of integration.

   (a)

   Find $\displaystyle\int6x^{2}(x^{3}-2)^{5}\,dx$.

   [1 mark]

   • A$\dfrac{(x^{3}-2)^{6}}{6}+C$
   • B$\dfrac{2x^{3}(x^{3}-2)^{6}}{6}+C$
   • C$2(x^{3}-2)^{6}+C$
   • D$\dfrac{(x^{3}-2)^{6}}{3}+C$

   (b)

   Find $\displaystyle\int4x\sin(x^{2})\,dx$.

   [1 mark]

   • A$-4\cos(x^{2})+C$
   • B$2\cos(x^{2})+C$
   • C$-2\cos(x^{2})+C$
   • D$-4x\cos(x^{2})+C$

   (c)

   Find $\displaystyle\int\frac{x}{x^{2}+5}\,dx$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A function $f$ is defined for $x>0$ and its derivative is $f'(x)=\dfrac{\ln x}{x}$. The graph of $y=f(x)$ passes through the point $(e,\,1)$.

   (a)

   By using the substitution $u=\ln x$, find $\displaystyle\int\frac{\ln x}{x}\,dx$.

   [3 marks]

   (b)

   Find $f(x)$, and hence find the exact value of $f(e^{3})$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   After heavy rain, water flows into a reservoir. The rate of flow, in thousands of m$^{3}$ per day, $t$ days after the rain starts is modelled by $R(t)=24te^{-0.2t^{2}}$, $t\ge0$. When $t=0$ the reservoir holds 500 thousand m$^{3}$ of water. Let $V(t)$ be the volume of water, in thousands of m$^{3}$, in the reservoir at time $t$. Assume no water leaves the reservoir.

   (a)

   (i) Find $\displaystyle\int24te^{-0.2t^{2}}\,dt$.
   (ii) Hence find an expression for $V(t)$.

   [6 marks]

   (b)

   (i) Explain why the volume of water in the reservoir never reaches 560 thousand m$^{3}$, and describe what happens to the volume in the long term.
   (ii) Find the exact time at which the reservoir holds 530 thousand m$^{3}$ of water.

   [6 marks]

   Total for question 4: 12 marks

End of questions