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

$$letf\left(x\right)={\int }_0^{+\mathrm{\infty }}\frac{dt}{t^4+x^4}withx>0$$$$1)determineaexplicitformoff\left(x\right)$$$$2)findalsog\left(x\right)={\int }_0^{\mathrm{\infty }}\frac{dt}{{(t^4+x^4)}^2}$$$$3)give{f}^{\left(n\right)}\left(x\right)atformofintegral$$$$4)calculate{\int }_0^{\mathrm{\infty }}\frac{dt}{t^4+8}and{\int }_0^{\mathrm{\infty }}\frac{dt}{{(t^4+8)}^2}$$$$5)calculate{A}_n={\int }_0^{\mathrm{\infty }}\frac{dt}{{(t^4+x^4)}^n}withnintegrnatural$$

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