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

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Question Number 131315 by Engr_Jidda last updated on 03/Feb/21

$$Giventhat{y}_n\left(t\right)={\varrho }^t{B}_{n}sin\frac{n\pi }{4}t$$$$Evaluate{\int }_0^4{\left[y_n\left(t\right)\right]}^{2}dt$$

Answered by Ar Brandon last updated on 03/Feb/21

$$\mathcal{I}={\int }_0^4{\varrho }^{2\mathrm{t}}{B}_{\mathrm{n}}{\sin }^2\frac{\mathrm{n}\pi }{4}\mathrm{t}=\frac{4B_{\mathrm{n}}}{\mathrm{n}\pi }{\int }_0^{\mathrm{n}\pi }{\mathrm{e}}^{\frac{8\mathrm{u}}{\mathrm{n}\pi }}{\sin }^2\mathrm{udu}$$$$=\frac{2B_{\mathrm{n}}}{\mathrm{n}\pi }{\int }_0^{\mathrm{n}\pi }{\mathrm{e}}^{\frac{8\mathrm{u}}{\mathrm{n}\pi }}(1-\cos 2\mathrm{u})\mathrm{du}$$

Commented by Ar Brandon last updated on 03/Feb/21

$$\mathrm{Applying}\mathrm{integration}\mathrm{by}-\mathrm{parts}$$$${\mathrm{we}}^{\prime }\mathrm{ll}\mathrm{get}\mathrm{result}.$$

Commented by Engr_Jidda last updated on 03/Feb/21

$$thankyousir$$

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