Reduction formulaeEdexcel International A Level Further Maths: Mind map

Exam questions on Reduction formulae

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 $n\ge2$.
   Hence find the exact value of $I_4$.2 marks
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 $n\ge2$.
   Find the exact area of $R$.2 marks
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.
   Show that $\sin(n+2)x-\sin nx=2\cos(n+1)x\sin x$.3 marks