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let-f-x-e-nx-ln-2-x-2-with-n-integr-natural-1-calculste-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-3-find-0-1-f-x-d-and-0-f-x-dx-




Question Number 84029 by mathmax by abdo last updated on 08/Mar/20
let f(x)=e^(−nx) ln(2+x^2 )   with n integr natural  1) calculste f^((n)) (x) and f^((n)) (0)  2) developp f at integr serie  3)find ∫_0 ^1  f(x)d and ∫_0 ^∞ f(x)dx
$${let}\:{f}\left({x}\right)={e}^{−{nx}} {ln}\left(\mathrm{2}+{x}^{\mathrm{2}} \right)\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculste}\:{f}^{\left({n}\right)} \left({x}\right)\:{and}\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie} \\ $$$$\left.\mathrm{3}\right){find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){d}\:{and}\:\int_{\mathrm{0}} ^{\infty} {f}\left({x}\right){dx} \\ $$

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