let-f-x-0-2pi-sin-2t-1-x-cos-t-dt-1-find-a-explicit-form-of-f-x-2-find-also-g-x-0-2pi-sin-2t-cost-1-xcost-2-dt-3-find-the-value-of-0-2pi-sin-2t-1-3-cos-t-dt

Tinku Tara June 4, 2023 Integration 0 Comments

Question Number 48264 by Abdo msup. last updated on 21/Nov/18

let f(x)=∫_0 ^(2π) ((sin(2t))/(1+x cos(t)))dt 1) find a explicit form of f(x) 2) find also g(x)=∫_0 ^(2π) ((sin(2t)cost)/((1+xcost)^2 ))dt 3)find the value of ∫_0 ^(2π) ((sin(2t))/(1+3 cos(t)))dt and ∫_0 ^(2π) ((cost sin(2t))/((1+3cost)^2 ))dt .

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{2}{t}\right)}{\mathrm{1}+{x}\:{cos}\left({t}\right)}{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{also}\:{g}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{sin}\left(\mathrm{2}{t}\right){cost}}{\left(\mathrm{1}+{xcost}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\frac{{sin}\left(\mathrm{2}{t}\right)}{\mathrm{1}+\mathrm{3}\:{cos}\left({t}\right)}{dt}\:{and} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\frac{{cost}\:{sin}\left(\mathrm{2}{t}\right)}{\left(\mathrm{1}+\mathrm{3}{cost}\right)^{\mathrm{2}} }{dt}\:. \\ $$

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let-f-x-0-2pi-sin-2t-1-x-cos-t-dt-1-find-a-explicit-form-of-f-x-2-find-also-g-x-0-2pi-sin-2t-cost-1-xcost-2-dt-3-find-the-value-of-0-2pi-sin-2t-1-3-cos-t-dt

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