Integrating standard functionsEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=6e^{2x}-\frac{3}{x}$ for $x>0$.

   (a)

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

   [1 mark]

   • A$3e^{2x}-3\ln x+c$
   • B$12e^{2x}-3\ln x+c$
   • C$6e^{2x}-3\ln x+c$
   • D$3e^{2x}+3\ln x+c$

   (b)

   Find the exact value of $\int_1^2 f(x)\,dx$.

   [1 mark]

   • A$3e^4-3e^2+3\ln2$
   • B$6e^4-6e^2-3\ln2$
   • C$3e^4-3e^2-3\ln2$
   • D$3e^4-3\ln2$

   (c)

   The curve $y=F(x)$ has gradient $f(x)$ and passes through the point $(1,3e^2+5)$. Find $F(x)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $g$ is defined by $g(x)=4\sin 3x+2\cos\frac{x}{2}$, where $x$ is in radians.

   (a)

   Find $\int g(x)\,dx$.

   [1 mark]

   • A$\frac43\cos3x+4\sin\frac{x}{2}+c$
   • B$-12\cos3x+\sin\frac{x}{2}+c$
   • C$-\frac43\cos3x+\sin\frac{x}{2}+c$
   • D$-\frac43\cos3x+4\sin\frac{x}{2}+c$

   (b)

   Find the exact value of $\int_0^{\pi/3}g(x)\,dx$.

   [1 mark]

   • A$-\frac23$
   • B$\frac{14}{3}$
   • C$\frac{11}{3}$
   • D$9$

   (c)

   Given that $\frac{dF}{dx}=g(x)$ and $F(0)=0$, find the exact value of $F(\pi)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves along a straight line. Its velocity is $v=3e^{-0.5t}+2$ m s$^{-1}$ at time $t$ seconds, for $t\geq0$, and its position relative to a fixed point $O$ is $s$ metres.

   (a)

   Find the displacement of the particle between $t=0$ and $t=4$, giving your answer to 3 significant figures.

   [3 marks]

   (b)

   Given that $s=5$ when $t=0$, find $s$ in terms of $t$, and hence find the exact value of $s$ when $t=2\ln3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A curve $C$ has gradient function $\frac{dy}{dx}=2^x+\frac{5}{2x}+\sin\frac{\pi x}{2}$ for $x>0$, and passes through the point $(1,3)$. A calculator may be used.

   (a)

   Find the equation of $C$. Hence find the value of $y$ when $x=2$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   (i) Find the exact value of $\int_1^3\frac{dy}{dx}\,dx$.
   (ii) Explain what this value represents for the curve $C$, and hence find the $y$-coordinate of $C$ when $x=3$, to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions