Question Number 37922 by gunawan last updated on 19/Jun/18 | ||
$${f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${g}\::\:\mathbb{N}\:\rightarrow\:\mathbb{R} \\ $$$${f}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{sin}\:{x}\:{dx} \\ $$$${g}\left({n}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} {x}^{{n}} \mathrm{cos}\:{x}\:{dx} \\ $$$$\frac{{f}\left({n}+\mathrm{1}\right)−{f}\left({n}\right)}{{g}\left({n}+\mathrm{1}\right)−{g}\left({n}\right)}=? \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 20/Jun/18 | ||
$${pls}\:{check}\:{the}\:{question}...{i}\:{tried}\:{but}\:{result} \\ $$$${not}\:{obtained} \\ $$ | ||
Commented by ajfour last updated on 20/Jun/18 | ||
$${yes},\:{the}\:{same}\:{here}. \\ $$ | ||
Answered by tanmay.chaudhury50@gmail.com last updated on 20/Jun/18 | ||
$$\int_{\mathrm{0}} ^{\mathrm{2}\Pi} {x}^{{n}} {sinx}\:{dx}={f}\left({n}\right) \\ $$$$=\mid{sinx}×\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}\mid_{\mathrm{0}} ^{\mathrm{2}\Pi} −\int_{\mathrm{0}} ^{\mathrm{2}\Pi} {cos}^{} {x}.\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}{dx} \\ $$$${f}\left({n}\right)=−\frac{\mathrm{1}}{{n}+\mathrm{1}}{g}\left({n}+\mathrm{1}\right)......\left(\mathrm{1}\right) \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}\Pi} {x}^{{n}} {cosxdx}={g}\left({n}\right) \\ $$$$=\mid{cosx}\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}\mid_{\mathrm{0}} ^{\mathrm{2}\Pi} +\int_{\mathrm{0}} ^{\mathrm{2}\Pi} {sinx}\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}{dx} \\ $$$$=\mid{cosx}.\frac{{x}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}\mid_{\mathrm{0}} ^{\mathrm{2}\Pi} +\frac{\mathrm{1}}{{n}+\mathrm{1}}{f}\left({n}+\mathrm{1}\right) \\ $$$$=\mathrm{1}.\frac{\left(\mathrm{2}\Pi\right)^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}+\frac{\mathrm{1}}{{n}+\mathrm{1}}{f}\left({n}+\mathrm{1}\right).....\left(\mathrm{2}\right) \\ $$$$\frac{{f}\left({n}+\mathrm{1}\right)−{f}\left({n}\right)}{{g}\left({n}+\mathrm{1}\right)−{g}\left({n}\right)} \\ $$$$\frac{\left({n}+\mathrm{1}\right){g}\left({n}\right)−\left(\mathrm{2}\Pi\right)^{{n}+\mathrm{1}} }{} \\ $$$${contd}\:...{wait}... \\ $$$$ \\ $$ | ||