Question Number 184753 by cortano1 last updated on 11/Jan/23
Answered by qaz last updated on 11/Jan/23
$$\underset{{r}\rightarrow\infty} {{lim}}\frac{{r}^{{c}} \int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}^{{r}} \mathrm{sin}\:{xdx}}{\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}^{{r}} \mathrm{cos}\:{xdx}}=\underset{{r}\rightarrow\infty} {{lim}}\frac{{r}^{{c}} \int_{\mathrm{0}} ^{\pi/\mathrm{2}} {e}^{{rln}\left(\frac{\pi}{\mathrm{2}}−{x}\right)} {cos}\:{xdx}}{\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {e}^{{rln}\left(\frac{\pi}{\mathrm{2}}−{x}\right)} {sin}\:{xdx}} \\ $$$$=\underset{{r}\rightarrow\infty} {{lim}}\frac{{r}^{{c}} \int_{\mathrm{0}} ^{\pi/\mathrm{2}} {e}^{{rln}\left(\mathrm{1}−\frac{\mathrm{2}}{\pi}{x}\right)} {cos}\:{xdx}}{\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {e}^{{rln}\left(\mathrm{1}−\frac{\mathrm{2}}{\pi}{x}\right)} {sin}\:{xdx}}=\underset{{r}\rightarrow\infty} {{lim}}\frac{{r}^{{c}} \int_{\mathrm{0}} ^{\infty} {e}^{−\frac{\mathrm{2}}{\pi}{rx}} {cos}\:{xdx}}{\int_{\mathrm{0}} ^{\infty} {e}^{−\frac{\mathrm{2}}{\pi}{rx}} \mathrm{sin}\:{xdx}}…….{by}\:{laplace}\:{method} \\ $$$$=\underset{{r}\rightarrow\infty} {{lim}}\frac{{r}^{{c}} \frac{\frac{\mathrm{2}}{\pi}{r}}{\left(\frac{\mathrm{2}}{\pi}{r}\right)^{\mathrm{2}} +\mathrm{1}}}{\frac{\mathrm{1}}{\left(\frac{\mathrm{2}}{\pi}{r}\right)^{\mathrm{2}} +\mathrm{1}}}=\underset{{r}\rightarrow\infty} {{lim}}\frac{\mathrm{2}{r}^{{c}+\mathrm{1}} }{\pi}={L}\:\:\:\:\:\Rightarrow{c}=−\mathrm{1}\:\:,\:{L}=\frac{\mathrm{2}}{\pi} \\ $$