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let-f-x-0-dt-t-4-x-4-with-x-gt-0-1-determine-a-explicit-form-of-f-x-2-find-also-g-x-0-dt-t-4-x-4-2-3-give-f-n-x-at-form-of-integral-4-calculate-0-

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Question Number 65767 by mathmax by abdo last updated on 03/Aug/19
let f(x) =∫_0 ^(+∞) (dt/(t^4 +x^4 )) with x>0 1) determine a explicit form of f(x) 2) find also g(x) =∫_0 ^∞ (dt/((t^4 +x^4 )^2 )) 3)give f^((n)) (x) at form of integral 4) calculate ∫_0 ^∞ (dt/(t^4 +8)) and ∫_0 ^∞ (dt/((t^4 +8)^2 )) 5) calculate A_n =∫_0 ^∞ (dt/((t^4 +x^4 )^n )) with n integr natural
$${let}\:\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{+\infty} \:\:\:\frac{{dt}}{{t}^{\mathrm{4}} +{x}^{\mathrm{4}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right){give}\:{f}^{\left({n}\right)} \left({x}\right)\:{at}\:{form}\:{of}\:{integral} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{{t}^{\mathrm{4}} \:+\mathrm{8}}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+\mathrm{8}\right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{5}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dt}}{\left({t}^{\mathrm{4}} \:+{x}^{\mathrm{4}} \right)^{{n}} }\:\:{with}\:{n}\:{integr}\:{natural} \\ $$

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