P3: IntegrationEdexcel International A Level Maths: Topic test

1. 1
   The function $f$ is defined by $f(x)=8e^{-2x}+3\sin\frac{x}{3}$ for $x\ge0$, where $x$ is in radians.

   (a)

   Which of the following is $\int f(x)\,dx$?

   [1 mark]

   • A$-4e^{-2x}+9\cos\frac{x}{3}+c$
   • B$-4e^{-2x}-9\cos\frac{x}{3}+c$
   • C$-16e^{-2x}-9\cos\frac{x}{3}+c$
   • D$-4e^{-2x}-\cos\frac{x}{3}+c$

   (b)

   Given that $F'(x)=f(x)$ and $F(0)=1$, find the exact value of $F(3\pi)$.

   [1 mark]

   • A$5-4e^{-6\pi}$
   • B$23-4e^{-2\pi}$
   • C$10-4e^{-6\pi}$
   • D$23-4e^{-6\pi}$

   (c)

   Find the exact value of $\displaystyle\int_{0}^{\frac{3\pi}{2}}3\sin\frac{x}{3}\,dx$, the integral of the second term of $f(x)$ over $0\le x\le\frac{3\pi}{2}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   During the first weeks of an appeal, a charity receives donations at a rate of $R=\dfrac{12}{t}+3^{t}$ thousand pounds per week, where $t\ge1$ is the time in weeks after the appeal began.

   (a)

   Which of the following is $\int R\,dt$?

   [1 mark]

   • A$12\ln t+\dfrac{3^{t}}{\ln3}+c$
   • B$12\ln t+3^{t}\ln3+c$
   • C$-\dfrac{12}{t^{2}}+\dfrac{3^{t}}{\ln3}+c$
   • D$12\ln t+\dfrac{3^{t+1}}{t+1}+c$

   (b)

   Find the total donations, in thousand pounds to 3 significant figures, received between $t=1$ and $t=2$.

   [1 mark]

   • A$14.3$
   • B$14.9$
   • C$13.8$
   • D$17.5$

   (c)

   Find the exact total donations, in thousand pounds, received between $t=1$ and $t=3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The functions $f$ and $g$ are defined by $f(x)=\dfrac{4x}{x^{2}+3}$ and $g(x)=\dfrac{3}{(3x-2)^{4}}$ for $x\ge1$.

   (a)

   Find the exact value of $\displaystyle\int_{1}^{3}f(x)\,dx$, giving your answer in the form $k\ln3$.

   [3 marks]

   (b)

   Find the exact value of $\displaystyle\int_{1}^{2}g(x)\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $f$ is defined by $f(x)=\sin^{2}x+\dfrac{2\cos x}{3+\sin x}$ for $0\le x\le\frac{\pi}{2}$.

   (a)

   (i) Use a double-angle identity to find $\int\sin^{2}x\,dx$.
   (ii) Find $\int\dfrac{2\cos x}{3+\sin x}\,dx$.
   (iii) Hence find the exact area of the region bounded by the curve $y=f(x)$, the $x$-axis, the $y$-axis and the line $x=\frac{\pi}{2}$.

   [6 marks]

   (b)

   (i) Show that the area of the region bounded by the curve $y=f(x)$, the $x$-axis, the $y$-axis and the line $x=\frac{\pi}{6}$ is $\frac{\pi}{12}-\frac{\sqrt3}{8}+2\ln\frac76$.
   (ii) Hence show that the area between the curve, the $x$-axis, the line $x=\frac{\pi}{6}$ and the line $x=\frac{\pi}{2}$ is $\frac{\pi}{6}+\frac{\sqrt3}{8}+2\ln\frac87$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $h$ is defined by $h(x)=\sec^{2}3x$ for $0\le x\le\frac{\pi}{12}$, where $x$ is in radians.

   (a)

   Which of the following is $\int h(x)\,dx$?

   [1 mark]

   • A$3\tan3x+c$
   • B$\tan3x+c$
   • C$\frac13\tan3x+c$
   • D$\frac13\sec3x+c$

   (b)

   Find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{12}}h(x)\,dx$.

   [1 mark]

   • A$\frac13$
   • B$3$
   • C$1$
   • D$\frac{\sqrt3}{9}$

   (c)

   Use $h(x)$ to find $\int\tan^{2}3x\,dx$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $p$ is defined by $p(x)=7^{x}-\dfrac{6}{x}+\cos4x$ for $x>0$, where $x$ is in radians.

   (a)

   Which of the following is $\int p(x)\,dx$?

   [1 mark]

   • A$7^{x}\ln7-6\ln x+\frac14\sin4x+c$
   • B$\dfrac{7^{x}}{\ln7}-6\ln x-\frac14\sin4x+c$
   • C$\dfrac{7^{x}}{\ln7}-6\ln x+4\sin4x+c$
   • D$\dfrac{7^{x}}{\ln7}-6\ln x+\frac14\sin4x+c$

   (b)

   Find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{8}}\cos4x\,dx$.

   [1 mark]

   • A$4$
   • B$\frac14$
   • C$1$
   • D$-\frac14$

   (c)

   Find the exact value of $\displaystyle\int_{1}^{2}\left(7^{x}-\frac{6}{x}\right)dx$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The functions $p$ and $q$ are defined by $p(x)=\dfrac{2x+3}{x^{2}+3x+1}$ and $q(x)=\tan^{2}\dfrac{x}{2}$ for $0\le x\le\frac{\pi}{2}$, where $x$ is in radians.

   (a)

   Show that $\displaystyle\int_{0}^{1}p(x)\,dx=\ln5$.

   [3 marks]

   (b)

   Find the exact value of $\displaystyle\int_{0}^{\frac{\pi}{2}}q(x)\,dx$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The power, in kilowatts, generated by two solar panels $A$ and $B$ at time $t$ hours after 06:00, for $0\le t\le12$, is modelled by $P_A=4\sin^{2}\dfrac{\pi t}{12}$ and $P_B=\dfrac{12}{2t+3}+\sin\dfrac{\pi t}{6}$. The energy generated, in kilowatt hours, over a period is the integral of the power over that period.

   (a)

   (i) Show that $P_A=2-2\cos\dfrac{\pi t}{6}$.
   (ii) Hence find the energy generated by panel $A$ between 06:00 and 09:00, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   (i) Find the exact energy generated by panel $B$ between 06:00 and 12:00.
   (ii) Find the energy generated by panel $A$ in the same period.
   (iii) Find, to 3 significant figures, how much more energy panel $B$ generates than panel $A$ in this period.

   [6 marks]

   Total for question 8: 12 marks

End of questions