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Question Number 131452 by Engr_Jidda last updated on 04/Feb/21
integrate ∫_0 ^(nπ) ϱ^((((8u)/(nπ)))) (1−cos2u)du
$${integrate} \\ $$$$\int_{\mathrm{0}} ^{{n}\pi} \varrho^{\left(\frac{\mathrm{8}{u}}{{n}\pi}\right)} \left(\mathrm{1}−{cos}\mathrm{2}{u}\right){du} \\ $$
Answered by bramlexs22 last updated on 04/Feb/21
hint : let ((8u)/(nπ)) = t ⇒ 2u=((tnπ)/4) ; du = ((nπ)/8) dt determinant (((u=nπ⇒t=8)),((u=0⇒t=0))) I=∫_0 ^( 8) e^t (1−cos ((tnπ)/4))(((nπ)/8)) dt using integration by parts
$${hint}\::\:{let}\:\frac{\mathrm{8}{u}}{{n}\pi}\:=\:{t}\:\Rightarrow\:\mathrm{2}{u}=\frac{{tn}\pi}{\mathrm{4}}\:;\:{du}\:=\:\frac{{n}\pi}{\mathrm{8}}\:{dt} \\ $$$$\:\begin{array}{|c|c|}{{u}={n}\pi\Rightarrow{t}=\mathrm{8}}\\{{u}=\mathrm{0}\Rightarrow{t}=\mathrm{0}}\\\hline\end{array} \\ $$$${I}=\int_{\mathrm{0}} ^{\:\mathrm{8}} {e}^{{t}} \left(\mathrm{1}−\mathrm{cos}\:\frac{{tn}\pi}{\mathrm{4}}\right)\left(\frac{{n}\pi}{\mathrm{8}}\right)\:{dt}\: \\ $$$$\:{using}\:{integration}\:{by}\:{parts} \\ $$

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