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Question Number 35678 by abdo imad last updated on 21/May/18

let f(t) =∫_0 ^∞ ((e^(−tx^2 ) arctan(x^2 ))/x^2 )dx with t>0 1) study the existencte of f(t) 2)calculate f^′ (t) 3)find a simple form of f(t).

$${let}\:{f}\left({t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:{with}\:{t}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{study}\:{the}\:{existencte}\:{of}\:{f}\left({t}\right) \\ $$ $$\left.\mathrm{2}\right){calculate}\:{f}^{'} \left({t}\right) \\ $$ $$\left.\mathrm{3}\right){find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({t}\right). \\ $$

Commented byprof Abdo imad last updated on 23/May/18

1) f(t) = ∫_0 ^1 ((e^(−tx^2 ) arctan(x^2 ))/x^2 )dx +∫_1 ^(+∞) ((e^(−tx^2 ) arctan(x^2 ))/x^2 )dx = I +J but we have ((e^(−tx^2 ) arctan(x^2 ))/x^2 )∼ e^(−tx^2 ) ⇒ lim_(x→0) ((e^(−tx^2 ) arctan(x^2 ))/x^2 ) = 1 so I converges also we have lim_(x→+∞) x^2 ((e^(−tx^2 ) arctan(x^2 ))/x^2 ) =0 the comvergence of J is assured so f(t)?exists for t>0 2) f^′ (t) = ∫_0 ^∞ (∂/∂t){ ((e^(−tx^2 ) arctan(x^2 ))/x^2 )}dx =−∫_0 ^∞ e^(−tx^2 ) arctan(x^2 )dx .

$$\left.\mathrm{1}\right)\:{f}\left({t}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\:+\int_{\mathrm{1}} ^{+\infty} \:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx} \\ $$ $$=\:{I}\:+{J}\:{but}\:{we}\:{have}\:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\sim\:{e}^{−{tx}^{\mathrm{2}} } \:\Rightarrow \\ $$ $${lim}_{{x}\rightarrow\mathrm{0}} \:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\:=\:\mathrm{1}\:{so}\:{I}\:{converges} \\ $$ $${also}\:{we}\:{have}\:{lim}_{{x}\rightarrow+\infty} \:\:{x}^{\mathrm{2}} \:\:\frac{{e}^{−{tx}^{\mathrm{2}} } {arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\:=\mathrm{0} \\ $$ $${the}\:{comvergence}\:{of}\:{J}\:{is}\:{assured}\:{so}\:{f}\left({t}\right)?{exists} \\ $$ $${for}\:{t}>\mathrm{0} \\ $$ $$\left.\mathrm{2}\right)\:{f}^{'} \left({t}\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{\partial}{\partial{t}}\left\{\:\:\frac{{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }\right\}{dx} \\ $$ $$=−\int_{\mathrm{0}} ^{\infty} \:\:\:{e}^{−{tx}^{\mathrm{2}} } \:{arctan}\left({x}^{\mathrm{2}} \right){dx}\:. \\ $$

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