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Category: Relation and Functions

f-continue-on-0-1-and-f-x-gt-0-on-0-1-prove-that-0-1-lnf-x-dx-ln-0-1-f-x-dx-

Question Number 97983 by abdomathmax last updated on 10/Jun/20 $$\mathrm{f}\:\mathrm{continue}\:\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right]\:\mathrm{and}\:\mathrm{f}\left(\mathrm{x}\right)>\mathrm{0}\:\mathrm{on}\:\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\mathrm{prove}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnf}\left(\mathrm{x}\right)\mathrm{dx}\leqslant\mathrm{ln}\left(\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\right) \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

prove-that-n-n-n-1-t-dt-2-prove-that-k-1-n-1-k-n-1-1-if-lt-1-and-k-1-n-1-k-ln-n-if-1-

Question Number 32369 by prof Abdo imad last updated on 23/Mar/18 $${prove}\:{that}\:\:{n}^{−\alpha} \:\sim\:\int_{{n}} ^{{n}+\mathrm{1}} \:{t}^{−\alpha} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}^{\alpha} }\:\sim\:\:\frac{{n}^{\mathrm{1}−\alpha} }{\mathrm{1}−\alpha}\:{if}\:\:\alpha<\mathrm{1}\:{and} \\ $$$$\sum_{{k}=\mathrm{1}} ^{{n}}…

1-let-n-Nand-A-n-0-pi-dx-1-cos-2-nx-calculate-A-n-2-f-C-0-0-pi-R-find-lim-n-0-pi-f-x-1-cos-2-nx-dx-

Question Number 32348 by abdo imad last updated on 23/Mar/18 $$\left.\mathrm{1}\right){let}\:{n}\:\in{Nand}\:\:\:{A}_{{n}} \:=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dx}}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}\:.{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{f}\in\:{C}^{\mathrm{0}} \left(\left[\mathrm{0},\pi\right],\:{R}\right)\:{find}\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{f}\left({x}\right)}{\mathrm{1}+{cos}^{\mathrm{2}} \left({nx}\right)}{dx}\:. \\ $$ Terms…