Reduction formulaeEdexcel A-Level Further Maths: Mind map

$\sin^nx$

$x^ne^x$

$x^n\sin x$

Reduction formulae

integrals with a parameter $n$

by partsbase caseexact values

$\tan^nx$

$\frac{\sin nx}{\sin x}$

Exam tips

Exam questions on Reduction formulae

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 $n\geq2$ .
Find the exact value of $I_5$ .2 marks
For integers $n\geq0$ , let $J_n=\int_0^1 x^n e^x\,dx$ .
Use the result of part (a) to find the exact value of $J_3$ .2 marks
For integers $n\geq0$ , let $T_n=\int_0^{\frac{\pi}{4}}\tan^n x\,dx$ .
Show that $T_n+T_{n-2}=\frac{1}{n-1}$ for $n\geq2$ .3 marks

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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).