Reduction formulaeEdexcel International A Level Further Maths: Flashcards

What these 13 flashcards ask

What is a reduction formula?
Reduction formula for In=\int0^{\frac{\pi}{2}}\sin^nx\,dx?
First step to derive it?
Choice of u and \frac{dv}{dx} in the derivation?
Why is the boundary term zero?
Identity used to remove \cos^2x?
Value of I0?
Value of I1?
Exact value of I4?
Exact value of I5?
Reduction formula for In=\int\frac{\sin nx}{\sin x}\,dx?
Key identity behind the \frac{\sin nx}{\sin x} formula?
Express \int0^{\frac{\pi}{2}}\sin^5x\cos^2x\,dx using In.

Exam questions on Reduction formulae

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
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
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

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).