Reduction formulaeEdexcel International A Level Further Maths: Subtopic test

1. 1
   Let $I_n=\int_0^{\frac{\pi}{2}}\sin^n x\,dx$ for integers $n\ge0$. It is given that $nI_n=(n-1)I_{n-2}$ for $n\ge2$.

   (a)

   Find the exact value of $I_2$.

   [1 mark]

   • A$\frac{\pi}{2}$
   • B$\frac{\pi}{4}$
   • C$\frac12$
   • D$\frac{\pi}{8}$

   (b)

   Find the exact value of $I_3$.

   [1 mark]

   • A$\frac23$
   • B$\frac32$
   • C$\frac{\pi}{3}$
   • D$\frac13$

   (c)

   Hence find the exact value of $I_4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The region $R$ lies between the curve $y=\sin^3x$, the $x$-axis and the line $x=\frac{\pi}{2}$, for $0\le x\le\frac{\pi}{2}$. Let $I_n=\int_0^{\frac{\pi}{2}}\sin^n x\,dx$, where $nI_n=(n-1)I_{n-2}$ for $n\ge2$.

   (a)

   $R$ is rotated through $2\pi$ radians about the $x$-axis. Which expression gives the volume of the solid formed?

   [1 mark]

   • A$\pi I_3$
   • B$\pi I_3^{\,2}$
   • C$\pi I_6$
   • D$2\pi I_6$

   (b)

   Find the exact value of $I_6$.

   [1 mark]

   • A$\frac{5}{16}$
   • B$\frac{3\pi}{16}$
   • C$\frac{9\pi}{40}$
   • D$\frac{5\pi}{32}$

   (c)

   Find the exact area of $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For integers $n\ge1$, let $I_n=\int\frac{\sin nx}{\sin x}\,dx$, where $\sin x\ne0$. Constants of integration may be ignored.

   (a)

   Show that $\sin(n+2)x-\sin nx=2\cos(n+1)x\sin x$.

   [3 marks]

   (b)

   Hence show that $I_{n+2}=I_n+\frac{2\sin(n+1)x}{n+1}$, and use it to find the exact value of $\int_0^{\frac{\pi}{2}}\frac{\sin5x}{\sin x}\,dx$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For integers $n\ge0$, let $I_n=\int_0^{\frac{\pi}{2}}\sin^nx\,dx$.

   (a)

   Prove that $nI_n=(n-1)I_{n-2}$ for $n\ge2$.

   [6 marks]

   (b)

   Use the result in (a) to find the exact value of $\int_0^{\frac{\pi}{2}}\sin^5x\cos^2x\,dx$.

   [6 marks]

   Total for question 4: 12 marks

End of questions