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let-f-x-1-3-arctan-x-x-t-dt-withx-gt-0-1-determine-a-explicit-form-of-f-x-2-give-f-x-at-form-of-integral-and-find-its-value-3-calculate-1-3-arctan-1-1-t-dt-and-1-3-ar




Question Number 58187 by maxmathsup by imad last updated on 19/Apr/19
let f(x) =∫_1 ^3  arctan(x+(x/t))dt   withx>0  1) determine a explicit form of f(x)  2)  give f^′ (x) at form of integral and find its value  3) calculate ∫_1 ^3  arctan(1+(1/t))dt   and    ∫_1 ^3  arctan(2+(2/t))dt .  4) calculate ∫_1 ^3   (2t−1)arctan(1+(1/t))dt .
$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left({x}+\frac{{x}}{{t}}\right){dt}\:\:\:{withx}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{give}\:{f}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{find}\:{its}\:{value} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:\:\:{and}\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{2}+\frac{\mathrm{2}}{{t}}\right){dt}\:. \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\left(\mathrm{2}{t}−\mathrm{1}\right){arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:. \\ $$

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