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Question Number 146621 by 777316 last updated on 14/Jul/21

$${\int }_0^{\mathrm{\infty }}\frac{xsin\left(4x\right)}{9+4x^2}dx=$$

Answered by mathmax by abdo last updated on 14/Jul/21

$$\mathrm{\Psi }={\int }_0^{\mathrm{\infty }}\frac{\mathrm{xsin}\left(4\mathrm{x}\right)}{{4\mathrm{x}}^2+9}\Rightarrow \mathrm{\Psi }{=}_{2\mathrm{x}=3\mathrm{z}}{\int }_0^{\mathrm{\infty }}\frac{3\mathrm{zsin}\left(4\frac{3\mathrm{z}}{2}\right)}{2.9(1+{\mathrm{z}}^2)}\frac{3}{2}\mathrm{dz}$$$$=\frac{1}{4}{\int }_0^{\mathrm{\infty }}\frac{\mathrm{zsin}\left(6\mathrm{z}\right)}{{\mathrm{z}}^2+1}\mathrm{dz}=\frac{1}{8}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{\mathrm{zsin}\left(6\mathrm{z}\right)}{{\mathrm{z}}^2+1}\mathrm{dz}=\frac{1}{8}\mathrm{Im}\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{{\mathrm{ze}}^{6\mathrm{iz}}}{{\mathrm{z}}^2)1}\mathrm{dz}\right)$$$$\mathrm{let}\phi \left(\mathrm{z}\right)=\frac{{\mathrm{ze}}^{6\mathrm{iz}}}{{\mathrm{z}}^2+1}\Rightarrow \phi \left(\mathrm{z}\right)=\frac{{\mathrm{ze}}^{6\mathrm{iz}}}{(\mathrm{z}-\mathrm{i})(\mathrm{z}+\mathrm{i})}$$$$\mathrm{residus}\mathrm{th}.\Rightarrow {\int }_{\mathrm{R}}\phi \left(\mathrm{z}\right)\mathrm{dz}=2\mathrm{i}\pi \mathrm{Res}(\phi ,\mathrm{i})$$$$\mathrm{Res}(\phi ,\mathrm{i})=\frac{{\mathrm{ie}}^{-6}}{2\mathrm{i}}\Rightarrow {\int }_{\mathrm{R}}\phi \left(\mathrm{z}\right)\mathrm{dz}=2\mathrm{i}\pi .\frac{{\mathrm{ie}}^{-6}}{2\mathrm{i}}=\frac{\mathrm{i}\pi }{{\mathrm{e}}^6}\Rightarrow$$$$\mathrm{\Psi }=\frac{1}{8}\times \frac{\pi }{{\mathrm{e}}^6}=\frac{\pi }{{8\mathrm{e}}^6}$$

Answered by KINMATICS last updated on 14/Jul/21

$$\int \frac{\mathrm{xsin}4\mathrm{x}}{9+{4\mathrm{x}}^2}\mathrm{dx}=\int \sin 4\mathrm{x}.\frac{\mathrm{x}}{{\left(2\mathrm{x}\right)}^2-{\left(3\mathrm{i}\right)}^2}\mathrm{dx}$$$$\int \frac{\mathrm{xsin}4\mathrm{x}}{9+{4\mathrm{x}}^2}\mathrm{dx}=\int \sin 4\mathrm{x}.\frac{1}{4}(\frac{1}{(2\mathrm{x}+3\mathrm{i})}+\frac{1}{2\mathrm{x}-3\mathrm{i}})\mathrm{dx}$$$$\int \frac{\mathrm{xsin}4\mathrm{x}}{9+{4\mathrm{x}}^2}\mathrm{dx}=\frac{1}{2}\int \frac{\sin 4\mathrm{x}}{(4\mathrm{x}+6\mathrm{i})}\mathrm{dx}+\frac{1}{2}\int \frac{\sin 4\mathrm{x}}{4\mathrm{x}-6\mathrm{i}}\mathrm{dx}$$$$\mathrm{For}\int \frac{\sin 4\mathrm{x}}{4\mathrm{x}+6\mathrm{i}}\mathrm{dx},\mathrm{let}4\mathrm{x}+6\mathrm{i}=\mathrm{u},4\mathrm{x}=\mathrm{u}-6\mathrm{i},\mathrm{dx}=\frac{\mathrm{du}}{4}$$$$\mathrm{for}\int \frac{\sin 4\mathrm{x}}{4\mathrm{x}-6\mathrm{i}}\mathrm{dx},\mathrm{let}4\mathrm{x}-6\mathrm{i}=\mathrm{y},4\mathrm{x}=\mathrm{y}+6\mathrm{i},\mathrm{dx}=\frac{\mathrm{dy}}{4}$$$$\Rightarrow \int \frac{\sin 4\mathrm{xdx}}{{4\mathrm{x}}^2+9}=\frac{1}{8}\int \frac{\sin (\mathrm{u}-6\mathrm{i})}{\mathrm{u}}\mathrm{du}+\frac{1}{8}\int \frac{\sin (\mathrm{y}+6\mathrm{i})}{\mathrm{y}}\mathrm{dy}$$$$$$$$\mathrm{I}=\frac{1}{8}\int \frac{\sin \left(\mathrm{u}\right)\cos \left(6\mathrm{i}\right)-\cos \left(\mathrm{u}\right)\sin \left(6\mathrm{i}\right)}{\mathrm{u}}\mathrm{du}+$$$$\frac{1}{8}\int \frac{\sin \left(\mathrm{y}\right)\cos \left(6\mathrm{i}\right)+\cos \left(\mathrm{y}\right)\sin \left(6\mathrm{i}\right)}{\mathrm{y}}\mathrm{dy}$$$$$$$$\mathrm{I}=\frac{\cos \left(6\mathrm{i}\right)}{8}\int \frac{\sin \left(\mathrm{u}\right)}{\mathrm{u}}\mathrm{du}-\frac{\sin \left(6\mathrm{i}\right)}{8}\int \frac{\cos \left(\mathrm{u}\right)}{\mathrm{u}}\mathrm{du}+$$$$\frac{\cos \left(6\mathrm{i}\right)}{8}\int \frac{\sin \left(\mathrm{y}\right)}{\mathrm{y}}\mathrm{dy}+\frac{\sin \left(6\mathrm{i}\right)}{8}\int \frac{\cos \left(\mathrm{y}\right)}{\mathrm{y}}\mathrm{dy}$$$$$$$$\mathrm{I}=\frac{\cos \left(6\mathrm{i}\right)}{8}\mathbf{S}\mathrm{i}\left(\mathrm{u}\right)-\frac{\sin \left(6\mathrm{i}\right)}{8}\mathrm{Ci}\left(\mathrm{u}\right)+$$$$\frac{\cos \left(6\mathrm{i}\right)}{8}\mathrm{Si}\left(\mathrm{y}\right)+\frac{\sin \left(6\mathrm{i}\right)}{8}\mathrm{Ci}\left(\mathrm{y}\right)+\mathrm{C}$$$$$$$$\mathrm{I}=\frac{\cos \left(6\mathrm{i}\right)}{8}\mathrm{Si}(4\mathrm{x}+6\mathrm{i})-\frac{\sin \left(6\mathrm{i}\right)}{8}\mathrm{Ci}(4\mathrm{x}+6\mathrm{i})$$$$\frac{\cos \left(6\mathrm{i}\right)}{8}\mathrm{Si}(4\mathrm{x}-6\mathrm{i})+\frac{\sin \left(6\mathrm{i}\right)}{8}\mathrm{Ci}(4\mathrm{x}-6\mathrm{i})+\mathrm{C}$$$$$$$$\mathrm{By}:\mathrm{Kin}\mathrm{Tom}$$$$\mathrm{contact}\mathrm{www}.\mathrm{hppt}://\mathrm{kintom}077\partial \mathrm{gmail}.\mathrm{com}$$

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