Integration (A2)AQA A-Level Maths: Topic test

1. 1
   Let $I=\displaystyle\int\left(\frac4x+10\sin\frac x5-e^{3x}\right)dx$ for $x>0$, where angles are in radians.

   (a)

   What is $\displaystyle\int10\sin\frac x5\,dx$?

   [1 mark]

   • A$50\cos\frac x5+c$
   • B$-2\cos\frac x5+c$
   • C$-50\cos\frac x5+c$
   • D$-10\cos\frac x5+c$

   (b)

   What is the value of $\displaystyle\int_0^{5\pi/2}10\sin\frac x5\,dx$?

   [1 mark]

   • A$50$
   • B$-50$
   • C$0$
   • D$10$

   (c)

   Find the exact value of $\displaystyle\int_1^2\left(\frac4x-e^{3x}\right)dx$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the limit $L=\displaystyle\lim_{\delta x\to0}\sum_{x=1}^{4}\frac{6}{x^2}\,\delta x$.

   (a)

   Which integral is equal to $L$?

   [1 mark]

   • A$\displaystyle\int_1^4\frac{6}{x}\,dx$
   • B$\displaystyle\int_1^4 6x^2\,dx$
   • C$\displaystyle\int_0^4\frac{6}{x^2}\,dx$
   • D$\displaystyle\int_1^4\frac{6}{x^2}\,dx$

   (b)

   What is the value of $L$?

   [1 mark]

   • A$\frac32$
   • B$\frac92$
   • C$-\frac92$
   • D$6\ln4$

   (c)

   A student approximates $L$ by $\sum\frac{6}{x^2}\times0.5$ using strips of width $0.5$ from $x=1$ to $x=4$, taking each height at the left-hand end of the strip. Explain why this approximation is greater than $L$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In each part use the substitution $u=1+e^x$, and give exact answers.

   (a)

   Show that $\displaystyle\int_0^{\ln3}\frac{e^x}{1+e^x}\,dx=\ln2$.

   [3 marks]

   (b)

   Find the exact value of $\displaystyle\int_0^{\ln3}\frac{e^{2x}}{1+e^x}\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Exact answers are required throughout this question, where $\ln$ denotes the natural logarithm.

   (a)

   Use integration by parts to find $\displaystyle\int_1^e x^2\ln x\,dx$.

   [6 marks]

   (b)

   Use integration by parts twice to find $\displaystyle\int_1^e x(\ln x)^2\,dx$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $f(x)=\dfrac{7x+4}{(x+2)(x-3)}$ for $x>3$.

   (a)

   Which expression is equal to $f(x)$ in partial fractions?

   [1 mark]

   • A$\dfrac{5}{x+2}+\dfrac{2}{x-3}$
   • B$\dfrac{2}{x+2}+\dfrac{5}{x-3}$
   • C$\dfrac{2}{x+2}-\dfrac{5}{x-3}$
   • D$\dfrac{7}{x+2}+\dfrac{4}{x-3}$

   (b)

   What is $\displaystyle\int f(x)\,dx$?

   [1 mark]

   • A$-\dfrac{2}{(x+2)^2}-\dfrac{5}{(x-3)^2}+c$
   • B$2\ln(x+2)-5\ln(x-3)+c$
   • C$\frac12\ln(x+2)+\frac15\ln(x-3)+c$
   • D$2\ln(x+2)+5\ln(x-3)+c$

   (c)

   Show that $\displaystyle\int_4^6f(x)\,dx=\ln432$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve satisfies the differential equation $\dfrac{dy}{dx}=\dfrac{x}{y^2}$, where $y>0$, and passes through the point $(0,2)$.

   (a)

   Which equation results from separating the variables?

   [1 mark]

   • A$\displaystyle\int y^2\,dy=\int x\,dx$
   • B$\displaystyle\int\frac{1}{y^2}\,dy=\int x\,dx$
   • C$\displaystyle\int y\,dy=\int x\,dx$
   • D$\displaystyle\int x\,dy=\int y^2\,dx$

   (b)

   What is the value of $y$ when $x=2$?

   [1 mark]

   • A$14$
   • B$\sqrt[3]{6}$
   • C$\sqrt[3]{14}$
   • D$\sqrt{14}$

   (c)

   Show that the solution can be written $y=\sqrt[3]{\dfrac{3x^2+16}{2}}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle moves along the $x$-axis. At time $t$ seconds its velocity is $v=9\sin3t+6e^{-2t}$ m s$^{-1}$, where $3t$ is in radians. When $t=0$ the particle is at the origin.

   (a)

   Find an expression for the displacement $x$ of the particle from the origin at time $t$.

   [3 marks]

   (b)

   Find the displacement of the particle from the origin when $t=\frac{\pi}{3}$, and the average velocity of the particle over the first $\frac{\pi}{3}$ seconds. Give your answers to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The number of fish $P$, in hundreds, in a lake $t$ years after it is stocked satisfies $\dfrac{dP}{dt}=\dfrac{P(5-P)}{10}$, where $0<P<5$ and $P=1$ when $t=0$.

   (a)

   By separating the variables and using partial fractions, show that $\ln\left(\dfrac{P}{5-P}\right)=\dfrac t2-\ln4$.

   [6 marks]

   (b)

   Show that $P=\dfrac{5}{1+4e^{-t/2}}$. Find the time at which $400$ fish are in the lake, and state what the model predicts about the number of fish in the long term.

   [6 marks]

   Total for question 8: 12 marks

End of questions