Question Number 133119 by abdomsup last updated on 19/Feb/21 | ||
$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{xsin}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{3}} }{dx} \\ $$ | ||
Answered by mindispower last updated on 19/Feb/21 | ||
$${let} \\ $$$${f}\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{xsin}\left(\mathrm{2}{x}\right)}{{x}^{\mathrm{2}} +{a}}{dx},\forall{a}\in\mathbb{R}_{+} −\left\{\mathrm{0}\right\} \\ $$$${f}\left({a}\right)=\frac{\mathrm{1}}{\mathrm{2}}{Im}\int_{−\infty} ^{\infty} \frac{{xe}^{\mathrm{2}{ix}} }{{x}^{\mathrm{2}} +{a}}{dx}.....{E} \\ $$$$\int_{−\infty} ^{\infty} \frac{{xe}^{\mathrm{2}{ix}} }{{x}^{\mathrm{2}} +{a}}=\mathrm{2}{i}\pi{Res}\left(\frac{{xe}^{\mathrm{2}{ix}} }{{x}^{\mathrm{2}} +{a}},{x}={i}\sqrt{{a}}\right) \\ $$$$=\mathrm{2}{i}\pi.\frac{{i}\sqrt{{a}}.{e}^{−\mathrm{2}\sqrt{{a}}} }{\mathrm{2}{i}\sqrt{{a}}}={i}\pi{e}^{−\mathrm{2}\sqrt{{a}}} \\ $$$${E}\Rightarrow{f}\left({a}\right)=\frac{\pi}{\mathrm{2}}{e}^{−\mathrm{2}\sqrt{{a}}} \\ $$$${f}'\left({a}\right)=\int_{\mathrm{0}} ^{\infty} \frac{−{xsin}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} +{a}\right)^{\mathrm{2}} }{dx} \\ $$$${f}''\left({a}\right)=\mathrm{2}\int\frac{{xsin}\left(\mathrm{2}{x}\right){dx}}{\left({x}^{\mathrm{2}} +{a}\right)^{\mathrm{3}} } \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{f}''\left(\mathrm{4}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{xsin}\left(\mathrm{2}{x}\right)}{\left({x}^{\mathrm{2}} +\mathrm{4}\right)^{\mathrm{2}} }{dx}...{our}\:{integral} \\ $$$${f}'\left({a}\right)=−\frac{\pi}{\mathrm{2}\sqrt{{a}}}{e}^{−\mathrm{2}\sqrt{{a}}} ,{f}''\left({a}\right)=\frac{\pi}{\mathrm{2}{a}}{e}^{−\mathrm{2}\sqrt{{a}}} +\frac{\pi}{\mathrm{4}{a}\sqrt{{a}}}{e}^{−\mathrm{2}\sqrt{{a}}} \\ $$$${f}''\left(\mathrm{4}\right)=\frac{\mathrm{5}\pi}{\mathrm{32}}{e}^{−\mathrm{4}} \\ $$ | ||
Answered by mathmax by abdo last updated on 20/Feb/21 | ||
$$\Phi=\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{xsin}\left(\mathrm{2x}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{3}} }\mathrm{dx}\:\Rightarrow\Phi=_{\mathrm{x}=\mathrm{2t}} \:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2tsin}\left(\mathrm{4t}\right)}{\mathrm{64}\left(\mathrm{t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\left(\mathrm{2dt}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{16}}\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{tsin}\left(\mathrm{4t}\right)}{\left(\mathrm{t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dt}\:=\frac{\mathrm{1}}{\mathrm{32}}\int_{−\infty} ^{+\infty} \:\frac{\mathrm{tsin}\left(\mathrm{4t}\right)}{\left(\mathrm{t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dt}\:=\frac{\mathrm{1}}{\mathrm{32}}\mathrm{Im}\left(\int_{−\infty} ^{+\infty} \:\frac{\mathrm{te}^{\mathrm{4it}} }{\left(\mathrm{t}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\mathrm{dt}\right) \\ $$$$\varphi\left(\mathrm{z}\right)=\frac{\mathrm{ze}^{\mathrm{4iz}} }{\left(\mathrm{z}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{3}} }\:\Rightarrow\varphi\left(\mathrm{z}\right)=\frac{\mathrm{ze}^{\mathrm{4iz}} }{\left(\mathrm{z}−\mathrm{i}\right)^{\mathrm{3}} \left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{3}} } \\ $$$$\int_{\mathrm{R}} \varphi\left(\mathrm{z}\right)\mathrm{dz}\:=\mathrm{2i}\pi\:\mathrm{Res}\left(\varphi,\mathrm{i}\right)\:\:\mathrm{and}\:\mathrm{Res}\left(\varphi,\mathrm{i}\right)=\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\frac{\mathrm{1}}{\left(\mathrm{3}−\mathrm{1}\right)!}\left\{\left(\mathrm{z}−\mathrm{i}\right)^{\mathrm{3}} \varphi\left(\mathrm{z}\right)\right\}^{\left(\mathrm{2}\right)} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\:\:\left\{\frac{\mathrm{ze}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{3}} }\right\}^{\left(\mathrm{2}\right)} \:\Rightarrow\mathrm{2Res}\left(\varphi,\mathrm{i}\right)=\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \left\{\:\:\frac{\left(\mathrm{e}^{\mathrm{4iz}} +\mathrm{4iz}\:\mathrm{e}^{\mathrm{4iz}} \right)\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{3}} −\mathrm{3}\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{2}} \mathrm{ze}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{6}} }\right\}^{\left(\mathrm{1}\right)} \\ $$$$\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\:\:\left\{\frac{\left(\mathrm{1}+\mathrm{4iz}\right)\mathrm{e}^{\mathrm{4iz}} \left(\mathrm{z}+\mathrm{i}\right)−\mathrm{3ze}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{4}} }\right\}^{\left(\mathrm{1}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\:\:\left\{\frac{\left.\left(\mathrm{1}+\mathrm{4iz}\right)\left(\mathrm{z}+\mathrm{i}\right)−\mathrm{3z}\right)\mathrm{e}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{4}} }\right\}^{\left(\mathrm{1}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\:\:\left\{\frac{\left(\mathrm{i}+\mathrm{4iz}^{\mathrm{2}} −\mathrm{3z}\right)\mathrm{e}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{4}} }\right\}^{\left(\mathrm{1}\right)} \\ $$$$=\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{i}} \:\:\:\:\frac{\left.\left(\mathrm{8iz}−\mathrm{3}\right)\mathrm{e}^{\mathrm{4iz}} \:+\mathrm{4i}\:\left(\mathrm{4iz}^{\mathrm{2}} −\mathrm{3z}+\mathrm{1}\right)\mathrm{e}^{\mathrm{4iz}} \right)\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{4}} −\mathrm{4}\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{3}} \left(\mathrm{4iz}^{\mathrm{2}} −\mathrm{3z}+\mathrm{i}\right)\mathrm{e}^{\mathrm{4iz}} }{\left(\mathrm{z}+\mathrm{i}\right)^{\mathrm{8}} } \\ $$$$....\mathrm{rest}\:\mathrm{to}\:\mathrm{finish}\:\mathrm{the}\:\mathrm{calculus}\:... \\ $$ | ||