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let-f-t-0-logx-x-2-t-2-dx-t-gt-0-1-calculate-f-n-t-and-f-n-0-2-developp-f-at-integr-serie-




Question Number 141933 by mathmax by abdo last updated on 24/May/21
let f(t) =∫_0 ^∞   ((logx)/(x^2  +t^2 ))dx   (t>0)  1)calculate f^((n)) (t)  and f^((n)) (0)  2) developp f at integr serie
$$\mathrm{let}\:\mathrm{f}\left(\mathrm{t}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{t}^{\mathrm{2}} }\mathrm{dx}\:\:\:\left(\mathrm{t}>\mathrm{0}\right) \\ $$$$\left.\mathrm{1}\right)\mathrm{calculate}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{t}\right)\:\:\mathrm{and}\:\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:\mathrm{developp}\:\mathrm{f}\:\mathrm{at}\:\mathrm{integr}\:\mathrm{serie} \\ $$
Commented by Mathspace last updated on 24/May/21
f^((n)) (1)
$${f}^{\left({n}\right)} \left(\mathrm{1}\right) \\ $$

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