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Category: Integration

Question-75248

Question Number 75248 by cesar.marval.larez@gmail.com last updated on 08/Dec/19 Commented by mathmax by abdo last updated on 14/Dec/19 $$\frac{{d}}{{dx}}\left(\frac{{e}^{{x}} }{\mathrm{1}+{x}}\right)\:=\frac{{e}^{{x}} \left(\mathrm{1}+{x}\right)−{e}^{{x}} ×\mathrm{1}}{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\:=\frac{\mathrm{xe}^{\mathrm{x}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\:\Rightarrow…

advanced-calculus-prove-that-0-x-2-cosh-2-x-2-dx-2-2-4-pi-1-2-

Question Number 140715 by mnjuly1970 last updated on 11/May/21 $$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:……{advanced}\:\:{calculus}…… \\ $$$$\:\:\:\:\:\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{{x}^{\mathrm{2}} }{{cosh}^{\mathrm{2}} \left({x}^{\mathrm{2}} \right)}{dx}\overset{?} {=}\frac{\sqrt{\mathrm{2}}\:−\mathrm{2}}{\mathrm{4}}\:\sqrt{\pi}\:\zeta\:\left(\:\frac{\mathrm{1}}{\mathrm{2}}\:\right)\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:………….. \\…