P4: IntegrationEdexcel International A Level Maths: Topic test

1. 1
   Let $I=\int_0^{\frac{\pi}{2}}\sin x\cos^{3}x\,\mathrm{d}x$.

   (a)

   Using the substitution $u=\cos x$, which of the following integrals is equal to $I$?

   [1 mark]

   • A$\int_1^0u^3\,\mathrm{d}u$
   • B$\int_0^{\frac{\pi}{2}}u^3\,\mathrm{d}u$
   • C$\int_0^1u^3\,\mathrm{d}u$
   • D$\int_0^1u^4\,\mathrm{d}u$

   (b)

   Find the value of $I$.

   [1 mark]

   • A$\frac14$
   • B$-\frac14$
   • C$\frac13$
   • D$\frac15$

   (c)

   Using the same substitution, find $\int\sin x\cos^{5}x\,\mathrm{d}x$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $J=\int x\sin2x\,\mathrm{d}x$.

   (a)

   Using integration by parts with $u=x$ and $\frac{\mathrm{d}v}{\mathrm{d}x}=\sin2x$, which expression is equal to $J$?

   [1 mark]

   • A$\frac12x\cos2x-\frac12\int\cos2x\,\mathrm{d}x$
   • B$-\frac12x\cos2x+\frac12\int\cos2x\,\mathrm{d}x$
   • C$-\frac12x\cos2x-\frac12\int\cos2x\,\mathrm{d}x$
   • D$-x\cos2x+\int\cos2x\,\mathrm{d}x$

   (b)

   Find the exact value of $\int_0^{\frac{\pi}{2}}x\sin2x\,\mathrm{d}x$.

   [1 mark]

   • A$-\frac{\pi}{4}$
   • B$\frac{\pi}{2}$
   • C$\frac{\pi}{8}$
   • D$\frac{\pi}{4}$

   (c)

   Hence find the exact value of $\int_0^{\frac{\pi}{2}}\left(3x\sin2x+4\right)\mathrm{d}x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\mathrm{f}(x)=\dfrac{5x+1}{(x-1)(2x+1)}$ for $x>1$.

   (a)

   Find constants $A$ and $B$ such that $\mathrm{f}(x)\equiv\dfrac{A}{x-1}+\dfrac{B}{2x+1}$.

   [3 marks]

   (b)

   Hence find the exact value of $\int_2^4\mathrm{f}(x)\,\mathrm{d}x$, giving your answer in the form $p\ln3+q\ln5$, where $p$ and $q$ are constants.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A population of bacteria, $P$ thousand, at time $t$ hours satisfies $\frac{\mathrm{d}P}{\mathrm{d}t}=\frac12P(5-P)$, where $0<P<5$. Initially $P=1$.

   (a)

   Solve the differential equation to show that $P=\dfrac{5}{1+4\mathrm{e}^{-\frac52t}}$.

   [6 marks]

   (b)

   (i) Find the time at which $P=4$, giving your answer to 3 significant figures.
   (ii) Using the substitution $u=4+\mathrm{e}^{\frac52t}$, find the exact value of $\int_0^2\dfrac{5}{1+4\mathrm{e}^{-\frac52t}}\,\mathrm{d}t$, giving your answer in the form $2\ln k$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The region $R$ is bounded by the curve $y=\dfrac{3}{x+1}$, the $x$-axis, the $y$-axis and the line $x=2$. The region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.

   (a)

   Which integral gives the volume of the solid?

   [1 mark]

   • A$\pi\int_0^2\frac{3}{(x+1)^2}\,\mathrm{d}x$
   • B$\pi\left(\int_0^2\frac{3}{x+1}\,\mathrm{d}x\right)^2$
   • C$2\pi\int_0^2\frac{3}{x+1}\,\mathrm{d}x$
   • D$\pi\int_0^2\frac{9}{(x+1)^2}\,\mathrm{d}x$

   (b)

   Find the exact volume of the solid.

   [1 mark]

   • A$9\pi\ln3$
   • B$6\pi$
   • C$2\pi$
   • D$6$

   (c)

   The region bounded by the same curve, the coordinate axes and the line $x=a$, where $a>0$, is rotated through $2\pi$ radians about the $x$-axis. Show that the volume of the solid formed is $\dfrac{9\pi a}{a+1}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A curve has parametric equations $x=3\cos\theta$, $y=2\sin\theta$, for $0\le\theta\le\pi$. The region $S$ is bounded by the curve and the $x$-axis.

   (a)

   Which integral gives the area of $S$?

   [1 mark]

   • A$\int_0^{\pi}6\sin^2\theta\,\mathrm{d}\theta$
   • B$-\int_0^{\pi}6\sin^2\theta\,\mathrm{d}\theta$
   • C$\int_0^{\pi}2\sin\theta\,\mathrm{d}\theta$
   • D$\int_0^{\pi}6\sin\theta\cos\theta\,\mathrm{d}\theta$

   (b)

   Find the area of $S$.

   [1 mark]

   • A$6\pi$
   • B$12$
   • C$3\pi$
   • D$\frac{3\pi}{2}$

   (c)

   Find the exact area of the part of $S$ for which $x\ge\frac32$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The mass $m$ kg of a chemical in a reactor at time $t$ hours satisfies $\frac{\mathrm{d}m}{\mathrm{d}t}=-km^{2}$, where $k$ is a positive constant. Initially $m=5$, and after 3 hours $m=2$.

   (a)

   Solve the differential equation to show that $\dfrac1m=kt+\dfrac15$.

   [3 marks]

   (b)

   Find the value of $k$, and hence find the time at which $m=0.5$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The region $R$ is bounded by the curve $y=\sqrt{x}\ln x$, the $x$-axis and the line $x=\mathrm{e}$, where $x\ge1$.

   (a)

   The region $R$ is rotated through $2\pi$ radians about the $x$-axis. Use integration by parts twice to show that the volume of the solid formed is $\frac{\pi}{4}\left(\mathrm{e}^2-1\right)$.

   [6 marks]

   (b)

   (i) Use the substitution $x=\mathrm{e}^{u}$ to show that the area of $R$ is $\int_0^1u\,\mathrm{e}^{\frac32u}\,\mathrm{d}u$.
   (ii) Hence find the exact area of $R$.

   [6 marks]

   Total for question 8: 12 marks

End of questions