Question Number 56698 by maxmathsup by imad last updated on 21/Mar/19
$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} \:+\mathrm{4}}{dx} \\ $$
Commented by maxmathsup by imad last updated on 23/Mar/19
$${let}\:{A}\:=\int_{\mathrm{0}} ^{\infty} \:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} \:+\mathrm{4}}\:{dx}\:\Rightarrow\mathrm{2}{A}\:=\:\int_{−\infty} ^{+\infty} \:\:\frac{{sin}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{4}} \:+\mathrm{4}}{dx}\:={Im}\:\left(\int_{−\infty} ^{+\infty} \:\frac{{e}^{{ix}^{\mathrm{2}} } }{{x}^{\mathrm{4}} \:+\mathrm{4}}{dx}\right)\:{let} \\ $$$${but}\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{{ix}^{\mathrm{2}} } }{{x}^{\mathrm{4}} \:+\mathrm{4}}\:{dx}\:=_{{x}=\sqrt{\mathrm{2}}{t}} \:\:\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{{i}\mathrm{2}{t}^{\mathrm{2}} } }{\mathrm{4}{t}^{\mathrm{4}} \:+\mathrm{4}}\:\sqrt{\mathrm{2}}{dt}\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{\mathrm{2}{it}^{\mathrm{2}} } }{{t}^{\mathrm{4}} \:+\mathrm{1}}\:{dt}\:{let} \\ $$$$\varphi\left({z}\right)\:=\:\frac{{e}^{\mathrm{2}{iz}^{\mathrm{2}} } }{{z}^{\mathrm{4}} \:+\mathrm{1}}\:\Rightarrow\varphi\left({z}\right)\:=\frac{{e}^{\mathrm{2}{iz}^{\mathrm{2}} } }{\left({z}^{\mathrm{2}} −{i}\right)\left({z}^{\mathrm{2}} \:+{i}\right)}\:=\frac{{e}^{\mathrm{2}{iz}^{\mathrm{2}} } }{\left({z}−{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)\left({z}+{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)\left({z}−{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)\left({z}+{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)} \\ $$$${so}\:{the}\:{poles}\:{of}\:\varphi\:{are}\:\overset{−} {+}{e}^{\frac{{i}\pi}{\mathrm{4}}} \:{and}\:\overset{−} {+}\:{e}^{−\frac{{i}\pi}{\mathrm{4}}} \:\:{residus}\:{theorem}\:{give} \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\left\{\:{Res}\left(\varphi,\:{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)+{Res}\left(\varphi,−{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)\right\} \\ $$$${Res}\left(\varphi,{e}^{\frac{{i}\pi}{\mathrm{4}}} \right)\:=\:\frac{{e}^{\mathrm{2}{i}\left({i}\right)} }{\mathrm{2}{e}^{\frac{{i}\pi}{\mathrm{4}}} \left(\mathrm{2}{i}\right)}\:=\frac{{e}^{−\mathrm{2}} }{\mathrm{4}{i}}\:{e}^{−\frac{{i}\pi}{\mathrm{4}}} \\ $$$${Res}\left(\varphi,−{e}^{−\frac{{i}\pi}{\mathrm{4}}} \right)\:=\:\frac{{e}^{\mathrm{2}{i}\left(−{i}\right)} }{−\mathrm{2}{e}^{−\frac{{i}\pi}{\mathrm{4}}} \left(−\mathrm{2}{i}\right)}\:=\frac{{e}^{\mathrm{2}} }{\mathrm{4}{i}}\:{e}^{\frac{{i}\pi}{\mathrm{4}}} \:\Rightarrow \\ $$$$\int_{−\infty} ^{+\infty} \:\varphi\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:\left\{\frac{{e}^{−\mathrm{2}} }{\mathrm{4}{i}}\:{e}^{−\frac{{i}\pi}{\mathrm{4}}} \:+\frac{{e}^{\mathrm{2}} }{\mathrm{4}{i}}\:{e}^{\frac{{i}\pi}{\mathrm{4}}} \right\}\:=\frac{\pi}{\mathrm{2}}\:\left\{\:{e}^{−\mathrm{2}} \left\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:−\frac{{i}}{\:\sqrt{\mathrm{2}}}\right\}\:+{e}^{\mathrm{2}} \left\{\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:+\frac{{i}}{\:\sqrt{\mathrm{2}}}\right\}\right\} \\ $$$$=\frac{\pi}{\mathrm{2}}\left\{\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\left({e}^{\mathrm{2}} \:+{e}^{−\mathrm{2}} \right)\:+\frac{{i}}{\:\sqrt{\mathrm{2}}}\left({e}^{\mathrm{2}} \:−{e}^{−\mathrm{2}} \right)\right\}\:\Rightarrow\:\mathrm{2}{A}\:=\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}}\left({e}^{\mathrm{2}} −{e}^{−\mathrm{2}} \right)\:\:\Rightarrow \\ $$$$\bigstar{A}\:=\frac{\pi}{\mathrm{4}\sqrt{\mathrm{2}}}\:\left({e}^{\mathrm{2}} \:−{e}^{−\mathrm{2}} \right)\:\bigstar \\ $$
Commented by maxmathsup by imad last updated on 23/Mar/19
$${error}\:{at}\:{final}\:{lines}\:\mathrm{2}{A}\:=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}}\:\frac{\pi}{\mathrm{2}\sqrt{\mathrm{2}}}\:\left({e}^{\mathrm{2}} −{e}^{−\mathrm{2}} \right)\:=\frac{\pi}{\mathrm{8}}\left({e}^{\mathrm{2}} −{e}^{−\mathrm{2}} \right)\:\Rightarrow \\ $$$${A}\:=\frac{\pi}{\mathrm{16}}\:\left({e}^{\mathrm{2}} \:−{e}^{−\mathrm{2}} \right)\:. \\ $$