let-W-x-arctan-xt-2-2-t-2-dt-1-find-a-explicit-form-of-f-x-2-find-the-value-of-t-2-2-t-2-1-x-2-t-4-dt-

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Question Number 48063 by maxmathsup by imad last updated on 18/Nov/18

let W(x) =∫_(−∞) ^(+∞) ((arctan(xt^2 ))/(2+t^2 ))dt 1) find a explicit form of f(x) 2) find the value of ∫_(−∞) ^(+∞) (t^2 /((2+t^2 )(1+x^2 t^4 )))dt .

$${let}\:{W}\left({x}\right)\:=\int_{−\infty} ^{+\infty} \:\:\frac{{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{2}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{t}^{\mathrm{2}} }{\left(\mathrm{2}+{t}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{\mathrm{2}} {t}^{\mathrm{4}} \right)}{dt}\:. \\ $$

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let-W-x-arctan-xt-2-2-t-2-dt-1-find-a-explicit-form-of-f-x-2-find-the-value-of-t-2-2-t-2-1-x-2-t-4-dt-

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