Reduction formulaeEdexcel A-Level Further Maths: Subtopic test

1. 1
   For integers $n\geq0$, let $I_n=\int_0^{\frac{\pi}{2}}\sin^n x\,dx$. It is known that $nI_n=(n-1)I_{n-2}$ for $n\geq2$.

   (a)

   Find the value of $I_1$.

   [1 mark]

   • A$0$
   • B$1$
   • C$\frac{\pi}{2}$
   • D$\frac12$

   (b)

   Express $I_4$ in terms of $I_2$.

   [1 mark]

   • A$\frac43I_2$
   • B$\frac14I_2$
   • C$\frac32I_2$
   • D$\frac34I_2$

   (c)

   Find the exact value of $I_5$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For integers $n\geq0$, let $J_n=\int_0^1 x^n e^x\,dx$.

   (a)

   Which expression gives $J_n$ in terms of $J_{n-1}$ for $n\geq1$?

   [1 mark]

   • A$e-nJ_{n-1}$
   • B$e+nJ_{n-1}$
   • C$nJ_{n-1}-e$
   • D$e-\frac{J_{n-1}}{n}$

   (b)

   Find the value of $J_1$.

   [1 mark]

   • A$e-1$
   • B$e$
   • C$1$
   • D$1-e$

   (c)

   Use the result of part (a) to find the exact value of $J_3$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   For integers $n\geq0$, let $T_n=\int_0^{\frac{\pi}{4}}\tan^n x\,dx$.

   (a)

   Show that $T_n+T_{n-2}=\frac{1}{n-1}$ for $n\geq2$.

   [3 marks]

   (b)

   Hence find the exact value of $T_4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For integers $n\geq0$, let $K_n=\int_0^{\frac{\pi}{2}}x^n\sin x\,dx$.

   (a)

   Show that $K_n=n\left(\frac{\pi}{2}\right)^{n-1}-n(n-1)K_{n-2}$ for $n\geq2$.

   [6 marks]

   (b)

   Hence find the exact value of $K_4$, giving your answer in the form $a\pi^3+b\pi+c$.

   [6 marks]

   Total for question 4: 12 marks

End of questions