$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left(\mathrm{3}{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\: \\$$
$${let}\:{A}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left(\mathrm{3}{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:\Rightarrow{A}\:=_{\sqrt{\mathrm{3}}{x}\:={t}} \:\:\:\mathrm{3}\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }\frac{{dt}}{\:\sqrt{\mathrm{3}}} \\$$$$=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({t}^{\mathrm{2}} \right)}{{t}^{\mathrm{2}} }{dt}\:\:{let}\:\varphi\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({t}^{\mathrm{2}} \right){e}^{−{xt}^{\mathrm{2}} } }{{t}^{\mathrm{2}} }{dt}\:\:{with}\:{x}\geqslant\mathrm{0} \\$$$$\varphi^{'} \left({x}\right)\:=−\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{xt}^{\mathrm{2}} } {sin}\left({t}^{\mathrm{2}} \right){dt}\:=−{Im}\left(\int_{\mathrm{0}} ^{\infty} \:{e}^{−{xt}^{\mathrm{2}} +{it}^{\mathrm{2}} } {dt}\right)\:{and} \\$$$$\int_{\mathrm{0}} ^{\infty} \:{e}^{\left(−{x}+{i}\right){t}^{\mathrm{2}} } {dt}\:=_{\sqrt{−{x}+{i}}{t}\:={u}} \:\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{u}^{\mathrm{2}} } \frac{{du}}{\:\sqrt{−{x}+{i}}} \\$$$$=\frac{\sqrt{\pi}}{\mathrm{2}\sqrt{−{x}+{i}}}\:\Rightarrow\varphi\left({x}\right)\:=\frac{\sqrt{\pi}}{\mathrm{2}}\int\:\frac{{dx}}{\:\sqrt{−{x}+{i}}}\:+{c} \\$$$$−{x}+{i}\:=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:{e}^{{iarctan}\left(−\frac{\mathrm{1}}{{x}}\right)} \:=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{e}^{−{i}\:{arctan}\left(\frac{\mathrm{1}}{{x}}\right)} \\$$$$=\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:{e}^{−{i}\left(\frac{\pi}{\mathrm{2}}−{arctanx}\right)} \:=−{i}\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }{e}^{{iatctanx}} \:\Rightarrow \\$$$$\sqrt{−{x}+{i}}\:=\sqrt{−{i}}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:{e}^{\frac{{i}}{\mathrm{2}}{arctan}\left({x}\right)} \:={e}^{−\frac{{i}\pi}{\mathrm{4}}} \:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{4}}} \:{e}^{\frac{{i}}{\mathrm{2}}{arctanx}} \:\Rightarrow \\$$$$\varphi\left({x}\right)\:=\frac{\sqrt{\pi}}{\mathrm{2}}\:{e}^{\frac{{i}\pi}{\mathrm{4}}} \:\:\int\:\:\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{−\frac{\mathrm{1}}{\mathrm{4}}} {e}^{−\frac{{i}}{\mathrm{2}}{arctanx}} {dx}\:+{c} \\$$$$….{be}\:{continued}…. \\$$