let-f-0-1-arctan-x-1-x-2-dx-with-real-1-find-f-interms-of-2-find-the-values-of-0-1-arctan-2x-1-2x-2-dx-and-0-1-arctan-4x-1-4x-2-dx-

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 57227 by maxmathsup by imad last updated on 31/Mar/19

let f(α)=∫_0 ^1 ((arctan(αx))/(1+αx^2 )) dx with α real 1) find f(α) interms of α 2) find the values of ∫_0 ^1 ((arctan(2x))/(1+2x^2 )) dx and ∫_0 ^1 ((arctan(4x))/(1+4x^2 ))dx

$${let}\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\alpha{x}\right)}{\mathrm{1}+\alpha{x}^{\mathrm{2}} }\:{dx}\:\:\:{with}\:\alpha\:{real} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left(\alpha\right)\:{interms}\:{of}\:\alpha \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left(\mathrm{2}{x}\right)}{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }\:{dx}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{arctan}\left(\mathrm{4}{x}\right)}{\mathrm{1}+\mathrm{4}{x}^{\mathrm{2}} }{dx} \\ $$

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let-f-0-1-arctan-x-1-x-2-dx-with-real-1-find-f-interms-of-2-find-the-values-of-0-1-arctan-2x-1-2x-2-dx-and-0-1-arctan-4x-1-4x-2-dx-

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