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Category: Integration

let-f-C-0-R-R-x-R-f-a-b-x-f-x-1-find-a-b-xf-x-dx-interms-of-a-b-f-x-dx-2-calculate-0-pi-xdx-1-sinx-

Question Number 50413 by Abdo msup. last updated on 16/Dec/18 $${let}\:{f}\:\in{C}^{\mathrm{0}} \left({R},{R}\right)\:/\:\forall\:{x}\in{R}\:\:{f}\left({a}+{b}−{x}\right)={f}\left({x}\right) \\ $$$$\left.\mathrm{1}\right)\:{find}\:\int_{{a}} ^{{b}} {xf}\left({x}\right){dx}\:{interms}\:{of}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{xdx}}{\mathrm{1}+{sinx}} \\ $$ Commented…

1-calculate-U-n-0-pi-dx-1-cos-2-nx-with-n-from-N-2-f-continue-from-0-pi-to-R-find-lim-n-0-pi-f-x-1-cos-2-nx-dx-

Question Number 50412 by Abdo msup. last updated on 16/Dec/18 $$\left.\mathrm{1}\right)\:{calculate}\:{U}_{{n}} =\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:{with}\:{n}\:{from}\:{N} \\ $$$$\left.\mathrm{2}\right)\:{f}\:{continue}\:{from}\:\left[\mathrm{0},\pi\right]\:{to}\:{R}\:\:{find} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx} \\ $$…

determine-f-C-0-0-1-R-verifying-0-1-f-x-dx-1-3-0-1-f-x-2-2-dx-

Question Number 50407 by Abdo msup. last updated on 16/Dec/18 $${determine}\:{f}\:\in{C}^{\mathrm{0}} \left(\left[\mathrm{0},\mathrm{1}\right],{R}\right)\:{verifying} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\frac{\mathrm{1}}{\mathrm{3}}\:+\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\left({f}\left({x}^{\mathrm{2}} \right)\right)^{\mathrm{2}} {dx} \\ $$ Terms of Service…