$${calculate}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{x}} }{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\$$
$${the}\:{Q}.{is}\:{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\left(−\mathrm{1}\right)^{{x}} }{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\$$
$${let}\:{I}\:=\int_{−\infty} ^{+\infty} \:\frac{\left(−\mathrm{1}\right)^{{x}} }{\left({x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} }{dx}\:\Rightarrow\:\:{I}\:=\:\int_{−\infty} ^{+\infty} \:\frac{{e}^{{i}\pi{x}} }{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} } \\$$$$=_{{x}=\sqrt{\mathrm{3}}{t}} \:\:\:\:\int_{−\infty} ^{+\infty} \:\frac{{e}^{{i}\pi\sqrt{\mathrm{3}}{t}} }{\mathrm{9}\left({t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\sqrt{\mathrm{3}}{dt}\:=\frac{\sqrt{\mathrm{3}}}{\mathrm{9}}\:\int_{−\infty} ^{+\infty} \:\frac{{e}^{{i}\pi\sqrt{\mathrm{3}}{t}} }{\left({t}^{\mathrm{2}} \:+\mathrm{1}\right)}{dt}\:{let} \\$$$${W}\left({z}\right)\:=\frac{{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} }{\left({z}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:\Rightarrow{W}\left({z}\right)\:=\frac{{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} }{\left({z}−{i}\right)^{\mathrm{2}} \left({z}+{i}\right)^{\mathrm{2}} }\:\:\:{residus}\:{theorem}\: \\$$$${give}\:\int_{−\infty} ^{+\infty} {W}\left({z}\right){dz}\:=\mathrm{2}{i}\pi{Res}\left({W},{i}\right) \\$$$${Res}\left({W},{i}\right)\:={lim}_{{z}\rightarrow{i}} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{2}−\mathrm{1}\right)!}\left\{\left({z}−{i}\right)^{\mathrm{2}} {W}\left({z}\right)\right\}^{\left(\mathrm{1}\right)} \\$$$$={lim}_{{z}\rightarrow{i}} \:\:\:\:\:\left\{\:\frac{{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} }{\left({z}+{i}\right)^{\mathrm{2}} }\right\}^{\left(\mathrm{1}\right)} \: \\$$$$={lim}_{{z}\rightarrow{i}} \:\:\:\:\frac{{i}\pi\sqrt{\mathrm{3}}{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} \left({z}+{i}\right)^{\mathrm{2}} −\mathrm{2}\left({z}+{i}\right){e}^{{i}\pi\sqrt{\mathrm{3}}{z}} }{\left({z}+{i}\right)^{\mathrm{4}} } \\$$$$={lim}_{{z}\rightarrow{i}} \:\:\:\:\frac{{i}\pi\sqrt{\mathrm{3}}{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} \left({z}+{i}\right)−\mathrm{2}{e}^{{i}\pi\sqrt{\mathrm{3}}{z}} }{\left({z}+{i}\right)^{\mathrm{3}} } \\$$$$=\frac{{i}\pi\sqrt{\mathrm{3}}{e}^{−\pi\sqrt{\mathrm{3}}} \left(\mathrm{2}{i}\right)−\mathrm{2}{e}^{−\pi\sqrt{\mathrm{3}}} }{\left(\mathrm{2}{i}\right)^{\mathrm{3}} }\:=\frac{\left(−\mathrm{2}\pi\sqrt{\mathrm{3}}−\mathrm{2}\right){e}^{−\pi\sqrt{\mathrm{3}}} }{−\mathrm{8}{i}}\:=\frac{\left(\mathrm{1}+\pi\sqrt{\mathrm{3}}\right){e}^{−\pi\sqrt{\mathrm{3}}} }{\mathrm{4}{i}} \\$$$$\Rightarrow\int_{−\infty} ^{+\infty} \:{W}\left({z}\right){dz}\:=\mathrm{2}{i}\pi×\frac{\left(\mathrm{1}+\pi\sqrt{\mathrm{3}}\right){e}^{−\pi\sqrt{\mathrm{3}}} }{\mathrm{4}{i}}\:=\frac{\pi}{\mathrm{2}}\left(\mathrm{1}+\pi\sqrt{\mathrm{3}}\right){e}^{−\pi\sqrt{\mathrm{3}}} \\$$$$\Rightarrow{I}\:=\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{18}}\left(\mathrm{1}+\pi\sqrt{\mathrm{3}}\right){e}^{−\pi\sqrt{\mathrm{3}}} \\$$