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f-N-R-g-N-R-f-n-0-2pi-x-n-sin-xdx-g-n-0-2pi-x-n-cos-xdx-f-n-1-f-n-g-n-1-g-n-




Question Number 949 by 123456 last updated on 04/May/15
f:N→R  g:N→R  f(n)=∫_0 ^(2π) x^n sin xdx  g(n)=∫_0 ^(2π) x^n cos xdx  ((f(n+1)−f(n))/(g(n+1)−g(n)))=?
$${f}:\mathbb{N}\rightarrow\mathbb{R} \\ $$$${g}:\mathbb{N}\rightarrow\mathbb{R} \\ $$$${f}\left({n}\right)=\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}{x}^{{n}} \mathrm{sin}\:{xdx} \\ $$$${g}\left({n}\right)=\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}{x}^{{n}} \mathrm{cos}\:{xdx} \\ $$$$\frac{{f}\left({n}+\mathrm{1}\right)−{f}\left({n}\right)}{{g}\left({n}+\mathrm{1}\right)−{g}\left({n}\right)}=? \\ $$

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