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Question Number 64993 by mathmax by abdo last updated on 23/Jul/19

let f(a) =∫_0 ^∞   ((cos(x^2 )−sin(x^2 ))/((x^2 +a^2 )^2 ))dx  with a>0  1) calculate f(a)  2) find the values of ∫_0 ^∞    ((cos(x^2 )−sin(x^2 ))/((x^2  +1)^2 )) and  ∫_0 ^∞   ((cos(x^2 )−sin(x^2 ))/((x^2  +3)^2 ))dx .

$${let}\:{f}\left({a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx}\:\:{with}\:{a}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{calculate}\:{f}\left({a}\right) \\ $$ $$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }\:{and} \\ $$ $$\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({x}^{\mathrm{2}} \right)−{sin}\left({x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{3}\right)^{\mathrm{2}} }{dx}\:. \\ $$

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