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Question Number 61646 by maxmathsup by imad last updated on 05/Jun/19

let f(x) =e^(−ax)  arctan(3x)   with a>0  1) calculate f^((n)) (x) and f^((n)) (0)  2) developp f (x) at integr serie .  3) calculate ∫_0 ^∞  f(x)dx .

$${let}\:{f}\left({x}\right)\:={e}^{−{ax}} \:{arctan}\left(\mathrm{3}{x}\right)\:\:\:{with}\:{a}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{calculate}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$ $$\left.\mathrm{2}\right)\:{developp}\:{f}\:\left({x}\right)\:{at}\:{integr}\:{serie}\:. \\ $$ $$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:{f}\left({x}\right){dx}\:. \\ $$

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