Question Number 60611 by aliesam last updated on 22/May/19
$${prove}\:{that}\:\underset{−\infty} {\overset{\infty} {\int}}{x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } {cos}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{\mathrm{2}} \right)\:{dx}\:=\:\frac{\sqrt{\pi}{sin}\left(\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)\right)}{\mathrm{4}\:\sqrt[{\mathrm{4}}]{\mathrm{125}}} \\ $$$${anyone}\:{can}\:{help}\:{me}\:{with}\:{this}\:{please} \\ $$$$ \\ $$
Answered by Smail last updated on 22/May/19
$${A}=\int_{−\infty} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } {cos}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{\mathrm{2}} \right){dx} \\ $$$$=\mathrm{2}\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } \frac{\mathrm{1}}{\mathrm{2}}{sin}\left(\mathrm{2}{x}^{\mathrm{2}} \right){dx} \\ $$$$={Im}\left(−\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } ×{e}^{−\mathrm{2}{ix}^{\mathrm{2}} } {dx}\right) \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} {dx}=\int_{\mathrm{0}} ^{\infty} {x}×{xe}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} {dx} \\ $$$${By}\:{parts} \\ $$$${u}={x}\Rightarrow{u}'=\mathrm{1} \\ $$$${v}'={xe}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} \Rightarrow{v}=\frac{−\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{i}+\mathrm{1}\right)}{e}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} {dx}=\frac{−\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{i}+\mathrm{1}\right)}\left[{xe}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} \right]_{\mathrm{0}} ^{\infty} +\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{i}+\mathrm{1}\right)}\int_{\mathrm{0}} ^{\infty} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} {dx} \\ $$$${let}\:\:{t}=\sqrt{\mathrm{2}{i}+\mathrm{1}}{x}\Rightarrow{dx}=\frac{{dt}}{\:\sqrt{\mathrm{2}{i}+\mathrm{1}}} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} \left(\mathrm{2}{i}+\mathrm{1}\right)} {dx}=\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{i}+\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\int_{\mathrm{0}} ^{\infty} {e}^{−{t}^{\mathrm{2}} } {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{2}{i}+\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }×\frac{\sqrt{\pi}}{\mathrm{2}}=\frac{\sqrt{\pi}}{\mathrm{4}\left(\sqrt{\mathrm{5}}\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}+{i}\frac{\mathrm{2}}{\:\sqrt{\mathrm{5}}}\right)\right)^{\mathrm{3}/\mathrm{2}} } \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}×\left(\sqrt{\mathrm{5}}\right)^{\mathrm{3}/\mathrm{2}} \left({e}^{{itan}^{−\mathrm{1}} \left(\mathrm{2}\right)} \right)^{\mathrm{3}/\mathrm{2}} }=\frac{\sqrt{\pi}}{\mathrm{4}×\mathrm{5}^{\mathrm{3}/\mathrm{4}} {e}^{{i}\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)} } \\ $$$$=\frac{\sqrt{\pi}}{\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}}{e}^{−{i}\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)} \\ $$$${Im}\left(−\frac{\sqrt{\pi}}{\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}}{e}^{−{i}\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)} \right)=\frac{\sqrt{\pi}}{\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}}{sin}\left(\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)\right) \\ $$$$\int_{−\infty} ^{\infty} {x}^{\mathrm{2}} {e}^{−{x}^{\mathrm{2}} } {cos}\left({x}^{\mathrm{2}} \right){sin}\left({x}^{\mathrm{2}} \right){dx}=\frac{\sqrt{\pi}}{\mathrm{4}\sqrt[{\mathrm{5}}]{\mathrm{125}}}{sin}\left(\frac{\mathrm{3}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left(\mathrm{2}\right)\right) \\ $$$$ \\ $$
Commented by aliesam last updated on 22/May/19
$${thank}\:{yoy}\:{verry}\:{much}\:{sir} \\ $$