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Category: Integration

1-let-f-R-C-2pi-periodic-even-f-x-x-x-0-pi-developp-f-at-fourier-serie-2-calculate-p-0-1-2p-1-2-

Question Number 33894 by math khazana by abdo last updated on 26/Apr/18 $$\left.\mathrm{1}\right){let}\:{f}\:\:{R}\rightarrow{C}\:\:\mathrm{2}\pi\:{periodic}\:{even}\:\:/{f}\left({x}\right)={x}\: \\ $$$$\forall\:{x}\in\left[\mathrm{0},\pi\left[\:\:{developp}\:{f}\:{at}\:{fourier}\:{serie}\right.\right. \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\sum_{{p}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{\left(\mathrm{2}{p}+\mathrm{1}\right)^{\mathrm{2}} }\:. \\ $$ Commented by abdo…

let-x-0-t-x-1-e-t-dt-with-x-gt-0-1-prove-that-x-1-x-pi-sin-pix-2-find-the-value-of-0-e-x-2-dx-

Question Number 33895 by math khazana by abdo last updated on 26/Apr/18 $${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt}\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\Gamma\left({x}\right)\Gamma\left(\mathrm{1}−{x}\right)=\:\frac{\pi}{{sin}\left(\pi{x}\right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}^{\mathrm{2}} } {dx}\:.…

let-F-x-0-pi-2-arctan-xtant-tant-dt-find-a-simple-form-of-f-x-2-find-the-value-of-0-pi-2-arctan-2tant-tant-dt-

Question Number 33884 by math khazana by abdo last updated on 26/Apr/18 $${let}\:{F}\left({x}\right)=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{arctan}\left({xtant}\right)}{{tant}}\:{dt}\:{find}\:{a}\:{simple} \\ $$$${form}\:{of}\:{f}\left({x}\right)\:. \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{arctan}\left(\mathrm{2}{tant}\right)}{{tant}}{dt}\:. \\ $$ Commented by…