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