Question Number 33979 by abdo imad last updated on 28/Apr/18 $${we}\:{give}\:{for}\:{t}>\mathrm{0}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sinx}}{{x}}\:{e}^{−{tx}} {dx}\:=\frac{\pi}{\mathrm{2}}\:−{arctant} \\ $$$${use}\:{this}\:{result}\:{to}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{1}−{e}^{−{x}} \right){sinx}}{{x}^{\mathrm{2}} }{dx}\:. \\ $$ Commented by abdo…
Question Number 99504 by pticantor last updated on 21/Jun/20 Answered by Ar Brandon last updated on 21/Jun/20 $$\mathrm{Posons}\:\mathrm{x}=\sqrt{\mathrm{2}}\mathrm{tan}\theta\Rightarrow\mathrm{dx}=\sqrt{\mathrm{2}}\mathrm{sec}^{\mathrm{2}} \theta\mathrm{d}\theta \\ $$$$\mathcal{I}=\int\frac{\sqrt{\mathrm{2}}\mathrm{sec}^{\mathrm{2}} \theta}{\left(\mathrm{2tan}^{\mathrm{2}} \theta+\mathrm{2}\right)^{\mathrm{2}} }\mathrm{d}\theta=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\int\frac{\mathrm{1}}{\mathrm{sec}^{\mathrm{2}} \theta}\mathrm{d}\theta=\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\int\mathrm{cos}^{\mathrm{2}}…
Question Number 99496 by ~blr237~ last updated on 21/Jun/20 $${convergence}\:{radius}\:{of}\:\:\underset{{n}\in\mathbb{N}} {\sum}\:\mathrm{2}^{{n}} {z}^{{n}!} \: \\ $$ Answered by mathmax by abdo last updated on 21/Jun/20 $$\mathrm{u}_{\mathrm{n}}…
Question Number 165010 by mnjuly1970 last updated on 24/Jan/22 $$ \\ $$$$\:\:\:\:\:\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\psi\:^{\left(\mathrm{2}\right)} \left(\:\mathrm{1}+\:{n}\:\right)}{\:{n}}\:=\:? \\ $$$$\:\:\:\:\:\:−−−\:{m}.{n}\:−−− \\ $$$$\:\:\:\: \\ $$ Terms of Service Privacy…
Question Number 33915 by prof Abdo imad last updated on 27/Apr/18 $${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} \:{dt}\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma^{\left({n}\right)} \left({x}\right)\:{with}\:{n}\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\Gamma\left({n}\:+\frac{\mathrm{3}}{\mathrm{2}}\right)\:{for}\:{n}\:{integr}. \\ $$ Commented by…
Question Number 164973 by mnjuly1970 last updated on 24/Jan/22 Answered by mr W last updated on 24/Jan/22 $${AC}^{\mathrm{2}} =\left(\mathrm{2}\sqrt{\mathrm{7}}\right)^{\mathrm{2}} +\left(\mathrm{3}\sqrt{\mathrm{7}}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{2}\sqrt{\mathrm{7}}\right)\left(\mathrm{3}\sqrt{\mathrm{7}}\right)\:\mathrm{cos}\:\mathrm{120}° \\ $$$${AC}^{\mathrm{2}} =\mathrm{28}+\mathrm{63}+\mathrm{42}=\mathrm{133} \\…
Question Number 164974 by mnjuly1970 last updated on 24/Jan/22 $$ \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{ln}\left(\:\mathrm{1}−\:{x}\:\right).\mathrm{ln}\left({x}\:\right)\:\:}{{x}^{\:\frac{\mathrm{3}}{\mathrm{2}}} }{dx}\overset{?} {=}\pi^{\:\mathrm{2}} −\mathrm{8ln}\left(\mathrm{2}\:\right)\: \\ $$$$\:\:\:\:\:−−−\:\:{m}.{n}\:−−− \\ $$$$ \\ $$ Answered by…
Question Number 33896 by math khazana by abdo last updated on 26/Apr/18 $${let}\:\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:{t}^{{x}−\mathrm{1}} \:{e}^{−{t}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\Gamma\left({x}+\mathrm{1}\right)\:{interms}\:{of}\:\Gamma\left({x}\right)\:\:\:{with}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\Gamma\left({n}\right)\:{for}\:{n}\:\in\:{N}^{\bigstar} \\ $$$$\left.\mathrm{3}\right){calculate}\:\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:. \\ $$…
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…
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}\:.…