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Question Number 70871 by Abdo msup. last updated on 09/Oct/19

$$calculatef\left(x\right)={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{cos\left(x(1+t^2)\right)}{1+t^2}dtwithx⩾0$$

Commented by mathmax by abdo last updated on 10/Oct/19

$$f\left(x\right)=Re\left({\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{e^{ix(1+t^2)}}{1+t^2}dt\right)letW\left(z\right)=\frac{e^{ix(1+z^2)}}{z^2+1}$$$$\Rightarrow W\left(z\right)=e^{ix}\frac{e^{{ixz}^2}}{(z-i)(z+i)}thepolesofWareiand-i$$$${\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}W\left(z\right)dz=2i\pi Res(W,i)=2i\pi e^{ix}\times \frac{e^{-ix}}{2i}=\pi \Rightarrow$$$$f\left(x\right)=\pi \mathrm{\forall }x$$$$$$

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