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let-f-x-0-cos-xcos-x-2-2-d-and-g-x-0-sin-xcos-x-2-2-d-1-find-a-explicit-form-of-f-x-and-g-x-2-find-the-value-of-0-cos-2cos-4-




Question Number 59160 by maxmathsup by imad last updated on 05/May/19
let f(x) =∫_0 ^∞   ((cos(xcosθ))/(x^2  +θ^2 )) dθ      and  g(x) =∫_0 ^∞      ((sin(xcosθ))/(x^2  +θ^2 )) dθ  1) find a explicit form of f(x) and g(x)  2) find the value of ∫_0 ^∞    ((cos(2cosθ))/(4+θ^2 )) dθ  and  ∫_0 ^∞    ((sin(2cosθ))/(4+θ^2 )) dθ  3) let u_n =f(n^2 )   study  the serie Σ u_n
$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta\:\:\:\:\:\:{and}\:\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{sin}\left({xcos}\theta\right)}{{x}^{\mathrm{2}} \:+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right)\:{and}\:{g}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta\:\:{and}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{sin}\left(\mathrm{2}{cos}\theta\right)}{\mathrm{4}+\theta^{\mathrm{2}} }\:{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{let}\:{u}_{{n}} ={f}\left({n}^{\mathrm{2}} \right)\:\:\:{study}\:\:{the}\:{serie}\:\Sigma\:{u}_{{n}} \\ $$

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