Question Number 70871 by Abdo msup. last updated on 09/Oct/19
$$\:{calculate}\:{f}\left({x}\right)=\int_{−\infty} ^{+\infty} \:\frac{{cos}\left({x}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\:{with}\:{x}\geqslant\mathrm{0} \\ $$
Commented by mathmax by abdo last updated on 10/Oct/19
$${f}\left({x}\right)\:={Re}\left(\:\int_{−\infty} ^{+\infty} \:\frac{{e}^{{ix}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)} }{\mathrm{1}+{t}^{\mathrm{2}} }{dt}\right)\:{let}\:{W}\left({z}\right)=\frac{{e}^{{ix}\left(\mathrm{1}+{z}^{\mathrm{2}} \right)} }{{z}^{\mathrm{2}} +\mathrm{1}} \\ $$$$\Rightarrow{W}\left({z}\right)={e}^{{ix}} \:\:\frac{{e}^{{ixz}^{\mathrm{2}} } }{\left({z}−{i}\right)\left({z}+{i}\right)}\:{the}\:{poles}\:{of}\:{W}\:{are}\:{i}\:{and}\:−{i} \\ $$$$\int_{−\infty} ^{+\infty} \:{W}\left({z}\right){dz}\:=\mathrm{2}{i}\pi\:{Res}\left({W},{i}\right)=\mathrm{2}{i}\pi\:{e}^{{ix}} \:×\frac{{e}^{−{ix}} }{\mathrm{2}{i}}\:=\pi\:\Rightarrow \\ $$$${f}\left({x}\right)=\pi\:\:\forall{x} \\ $$$$ \\ $$