Which method derives nIn=(n-1)I{n-2}?

Integration by parts with u=\sin^{n-1}x, v'=\sin x, then \cos^2x=1-\sin^2x.

Why does \int0^{\frac{\pi}{2}}x^n\sin x\,dx need integration by parts twice?

One application turns \sin x into \cos x; a second turns it back to \sin x with x^{n-2}.

Result for Kn=\int0^{\frac{\pi}{2}}x^n\sin x\,dx?

Kn=n\left(\frac{\pi}{2}\right)^{n-1}-n(n-1)K{n-2}, n\geq2.

Identity behind the reduction for \int\frac{\sin nx}{\sin x}\,dx?

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

Reduction formula for In=\int\frac{\sin nx}{\sin x}\,dx?

I{n+2}=In+\frac{2\sin((n+1)x)}{n+1} (plus a constant).

Reduction formulaeEdexcel A-Level Further Maths: Flashcards

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State the reduction formula for $I_n=\int_0^{\frac{\pi}{2}}\sin^nx\,dx$.

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State the reduction formula for $I_n=\int_0^{\frac{\pi}{2}}\sin^nx\,dx$ .: $nI_n=(n-1)I_{n-2}$ for $n\geq2$ .
Values of $I_0$ and $I_1$ for $\int_0^{\frac{\pi}{2}}\sin^nx\,dx$ ?: $I_0=\frac{\pi}{2}$ and $I_1=1$ .
Which method derives $nI_n=(n-1)I_{n-2}$ ?: Integration by parts with $u=\sin^{n-1}x$ , $v'=\sin x$ , then $\cos^2x=1-\sin^2x$ .
Find $\int_0^{\frac{\pi}{2}}\sin^3x\,dx$ .: $\frac23$ (from $3I_3=2I_1$ ).
Find $\int_0^{\frac{\pi}{2}}\sin^4x\,dx$ .: $\frac{3\pi}{16}$ .
Reduction formula for $J_n=\int_0^1x^ne^x\,dx$ ?: $J_n=e-nJ_{n-1}$ .
What is $J_0$ and hence $J_1$ ?: $J_0=e-1$ , $J_1=1$ .
Why does $\int_0^{\frac{\pi}{2}}x^n\sin x\,dx$ need integration by parts twice?: One application turns $\sin x$ into $\cos x$ ; a second turns it back to $\sin x$ with $x^{n-2}$ .
Result for $K_n=\int_0^{\frac{\pi}{2}}x^n\sin x\,dx$ ?: $K_n=n\left(\frac{\pi}{2}\right)^{n-1}-n(n-1)K_{n-2}$ , $n\geq2$ .
Relation between $T_n$ and $T_{n-2}$ for $T_n=\int_0^{\frac{\pi}{4}}\tan^nx\,dx$ ?: $T_n+T_{n-2}=\frac{1}{n-1}$ .
Identity behind the reduction for $\int\frac{\sin nx}{\sin x}\,dx$ ?: $\sin(n+2)x-\sin nx=2\cos(n+1)x\sin x$ .
Reduction formula for $I_n=\int\frac{\sin nx}{\sin x}\,dx$ ?: $I_{n+2}=I_n+\frac{2\sin((n+1)x)}{n+1}$ (plus a constant).

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).