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Category: Integration

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

Question Number 64160 by mathmax by abdo last updated on 14/Jul/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{{x}\:+\mathrm{2}^{{t}} }\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{determine}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left({x}+\mathrm{2}^{{x}} \right)^{\mathrm{2}} } \\ $$$$\left.\mathrm{3}\right)\:{give}\:{f}^{\left({n}\right)}…