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Category: Integration

let-f-C-0-0-pi-R-prove-that-lim-n-0-pi-f-x-sin-nx-dx-2-pi-0-pi-f-x-dx-

Question Number 36932 by maxmathsup by imad last updated on 07/Jun/18 $${let}\:{f}\:\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],{R}\right)\:\:{prove}\:{that} \\ $$$${lim}_{{n}\rightarrow+\infty} \:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right)\:\mid{sin}\left({nx}\right)\mid{dx}\:=\frac{\mathrm{2}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {f}\left({x}\right){dx}\:. \\ $$ Terms of Service…

Calculate-J-dx-x-x-2-x-1-2-K-x-3-x-1-x-2-2-2-dx-L-dx-x-x-2-1-

Question Number 102462 by Ar Brandon last updated on 09/Jul/20 $$\mathrm{Calculate}\:; \\ $$$$\mathrm{J}=\int\frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\mathrm{K}=\int\frac{\mathrm{x}^{\mathrm{3}} +\mathrm{x}−\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$$$\mathrm{L}=\int\frac{\mathrm{dx}}{\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}} \\ $$…

3-cos-x-3-cos-x-dx-

Question Number 167989 by cortano1 last updated on 31/Mar/22 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{\mathrm{3}−\mathrm{cos}\:{x}}{\mathrm{3}+\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$ Answered by nimnim last updated on 31/Mar/22 $$\:\:\mathrm{I}=\int\:\frac{\mathrm{6}−\mathrm{3}−\mathrm{cosx}}{\mathrm{3}+\mathrm{cosx}}\mathrm{dx}=\int\left(\frac{\mathrm{6}}{\mathrm{3}+\mathrm{cosx}}−\mathrm{1}\right)\mathrm{dx} \\ $$$$\:\:\:\:=\mathrm{6}\int\frac{\mathrm{dx}}{\mathrm{3}+\mathrm{cosx}}−\int\mathrm{dx}=\mathrm{I}_{\mathrm{1}} −\mathrm{x} \\ $$$$\:\:\:\:\mathrm{put}\:\mathrm{t}=\mathrm{tan}\left(\frac{\mathrm{x}}{\mathrm{2}}\right)\Rightarrow\:\frac{\mathrm{2}}{\mathrm{1}+\mathrm{t}^{\mathrm{2}}…