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Question Number 74800 by mathmax by abdo last updated on 30/Nov/19
$${study}\:{the}\:{existence}\:{of}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{tcos}\left({tx}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$
Answered by mind is power last updated on 30/Nov/19
![x=0 ∫_0 ^(+∞) (t/(1+t^2 ))dt divege for x#0 ipp [_0 ^(+∞) (t/(t^2 +1)).((sin(xt))/x)]−(1/x)∫_0 ^(+∞) sin(xt).(((1−t^2 ))/((1+t^2 )^2 ))dt =−(1/x)∫_0 ^(+∞) (((1−t^2 )sin(xt))/((1+t^2 )^2 ))dt exist cv abdolutly ≤(1/(∣x∣))∫_0 ^(+∞) ((∣(1−t^2 )∣)/((1+t^2 )^2 ))dt exist since at[+∞ (((t^2 −1))/((t^2 +1)^2 ))∼(1/t^2 ),wich is Reimann/integrabl at +∞ f(x) exist overR^∗](Q74812.png)
$$\mathrm{x}=\mathrm{0} \\ $$$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{t}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}} }\mathrm{dt}\:\:\mathrm{divege} \\ $$$$\mathrm{for}\:\mathrm{x}#\mathrm{0} \\ $$$$\mathrm{ipp}\:\:\:\:\:\left[_{\mathrm{0}} ^{+\infty} \frac{\mathrm{t}}{\mathrm{t}^{\mathrm{2}} +\mathrm{1}}.\frac{\mathrm{sin}\left(\mathrm{xt}\right)}{\mathrm{x}}\right]−\frac{\mathrm{1}}{\mathrm{x}}\int_{\mathrm{0}} ^{+\infty} \mathrm{sin}\left(\mathrm{xt}\right).\frac{\left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dt} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{x}}\int_{\mathrm{0}} ^{+\infty} \frac{\left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)\mathrm{sin}\left(\mathrm{xt}\right)}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dt} \\ $$$$\mathrm{exist}\:\mathrm{cv}\:\mathrm{abdolutly} \\ $$$$\leqslant\frac{\mathrm{1}}{\mid\mathrm{x}\mid}\int_{\mathrm{0}} ^{+\infty} \frac{\mid\left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)\mid}{\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dt} \\ $$$$\mathrm{exist}\:\mathrm{since}\:\:\:\mathrm{at}\left[+\infty\:\right. \\ $$$$\frac{\left(\mathrm{t}^{\mathrm{2}} −\mathrm{1}\right)}{\left(\mathrm{t}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }\sim\frac{\mathrm{1}}{\mathrm{t}^{\mathrm{2}} },\mathrm{wich}\:\mathrm{is}\:\mathrm{Reimann}/\mathrm{integrabl}\:\mathrm{at}\:+\infty \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{exist}\:\mathrm{over}\mathbb{R}^{\ast} \\ $$