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Category: Integration

prove-that-1-e-0-1-e-x-x-2-dx-1-

Question Number 30442 by abdo imad last updated on 22/Feb/18 $${prove}\:{that}\:\:\:\frac{\mathrm{1}}{{e}}\:\leqslant\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{−\left({x}−\left[{x}\right]\right)^{\mathrm{2}} } \:{dx}\leqslant\mathrm{1}. \\ $$ Commented by alex041103 last updated on 22/Feb/18 $${You}\:{can}\:{see}\:{that}\:{if}\:{x}\in\left[\mathrm{0},\mathrm{1}\right),\:{then}…

f-is-a-integrable-function-wich-verify-f-x-pi-f-x-prove-that-0-f-x-sinx-x-dx-0-pi-2-f-x-dx-

Question Number 95943 by mathmax by abdo last updated on 28/May/20 $$\mathrm{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{integrable}\:\mathrm{function}\:\mathrm{wich}\:\mathrm{verify}\:\mathrm{f}\left(\mathrm{x}+\pi\right)=\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)×\frac{\mathrm{sinx}}{\mathrm{x}}\mathrm{dx}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$ Terms of Service Privacy Policy…

Question-161443

Question Number 161443 by smallEinstein last updated on 18/Dec/21 Answered by mathmax by abdo last updated on 18/Dec/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{log}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\mathrm{letf}\left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{a}}…