# Given-that-y-n-t-t-B-n-sin-npi-4-t-Evaluate-0-4-y-n-t-2-dt-

Question Number 131315 by Engr_Jidda last updated on 03/Feb/21
$${Given}\:{that}\:{y}_{{n}} \left({t}\right)=\varrho^{{t}} {B}_{{n}} {sin}\frac{{n}\pi}{\mathrm{4}}{t} \\$$$${Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{4}} \left[{y}_{{n}} \left({t}\right)\right]^{\mathrm{2}} {dt} \\$$
Answered by Ar Brandon last updated on 03/Feb/21
$$\mathcal{I}=\int_{\mathrm{0}} ^{\mathrm{4}} \varrho^{\mathrm{2t}} {B}_{\mathrm{n}} \mathrm{sin}^{\mathrm{2}} \frac{\mathrm{n}\pi}{\mathrm{4}}\mathrm{t}=\frac{\mathrm{4}{B}_{\mathrm{n}} }{\mathrm{n}\pi}\int_{\mathrm{0}} ^{\mathrm{n}\pi} \mathrm{e}^{\frac{\mathrm{8u}}{\mathrm{n}\pi}} \mathrm{sin}^{\mathrm{2}} \mathrm{udu} \\$$$$\:\:\:=\frac{\mathrm{2}{B}_{\mathrm{n}} }{\mathrm{n}\pi}\int_{\mathrm{0}} ^{\mathrm{n}\pi} \mathrm{e}^{\frac{\mathrm{8u}}{\mathrm{n}\pi}} \left(\mathrm{1}−\mathrm{cos2u}\right)\mathrm{du} \\$$
Commented by Ar Brandon last updated on 03/Feb/21
$$\mathrm{Applying}\:\mathrm{integration}\:\mathrm{by}-\mathrm{parts}\: \\$$$$\mathrm{we}'\mathrm{ll}\:\mathrm{get}\:\mathrm{result}. \\$$
Commented by Engr_Jidda last updated on 03/Feb/21
$${thank}\:{you}\:{sir} \\$$