Question Number 164419 by akornes last updated on 16/Jan/22 $${please}\:{help}\:{me} \\ $$$${prouve}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{lnt}}{{t}^{\mathrm{2}} −\mathrm{1}}{dt}=\frac{\pi^{\mathrm{2}} }{\mathrm{8}} \\ $$ Answered by Ar Brandon last updated on…
Question Number 33346 by prof Abdo imad last updated on 14/Apr/18 $${let}\:{a}\:{and}\:{b}\:{from}\:{R}\:/{a}<{b}\:\:{f}\:\:\left[{a},{b}\right]\rightarrow{C}\:{continje} \\ $$$${prove}\:{that}\:\:\forall{n}\:\in{N}\:\:\int_{{a}} ^{{b}} \left(\prod_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left({x}+{k}\right)\right){f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\Rightarrow\:{f}=\mathrm{0} \\ $$ Terms of Service…
Question Number 98883 by mathmax by abdo last updated on 16/Jun/20 $$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\mathrm{x}^{\mathrm{8}} \:+\mathrm{x}^{\mathrm{4}} +\mathrm{1}} \\ $$ Answered by maths mind last updated on…
Question Number 33345 by prof Abdo imad last updated on 14/Apr/18 $$\left.{for}\:{x}\in\right]\mathrm{0},+\infty\left[\:{let}\:\psi\left({x}\right)\:=\:\frac{\Gamma^{'} \left({x}\right)}{\Gamma\left({x}\right)}\right. \\ $$$$\left.\mathrm{1}\right){prove}\:{that}\:\psi\left({x}\right)\:=−\frac{\mathrm{1}}{{x}}\:−\gamma\:+{x}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{{n}\left({x}+{n}\right)} \\ $$$$\left.\mathrm{2}\right){ptove}\:{that}\:\gamma\:=−\Gamma^{'} \left(\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−{x}} {ln}\left({x}\right){dx}\:=−\gamma\:.…
Question Number 33343 by prof Abdo imad last updated on 14/Apr/18 $$\left.{prove}\:{that}\:\forall\:\alpha\:\in\right]\mathrm{1},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}+\frac{{x}}{{n}}\right)^{{n}} \:{e}^{−\alpha{x}} {dx}\:=\:\frac{\mathrm{1}}{\alpha−\mathrm{1}}\:. \\ $$ Commented by abdo imad…
Question Number 33344 by prof Abdo imad last updated on 14/Apr/18 $$\left.{prove}\:{that}\:\:\forall\:\alpha\:\in\right]\mathrm{0},+\infty\left[\right. \\ $$$${lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \:{x}^{\alpha−\mathrm{1}} {dx}\:=\Gamma\left(\alpha\right)\:. \\ $$ Commented by prof Abdo…
Question Number 33341 by prof Abdo imad last updated on 14/Apr/18 $${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left({a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \right)} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of} \\ $$$${A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\:{x}^{\mathrm{2}} \:+\mathrm{1}\:−{sin}^{\mathrm{2}}…
Question Number 33342 by prof Abdo imad last updated on 14/Apr/18 $${let}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} \right){dx}\:{and} \\ $$$${J}_{{n}} \:=\:\int_{\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{x}}\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {dx}\:\:,{n}\:{integr}\:{not}\:\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{lim}_{} \:{I}_{{n}}…
Question Number 33339 by prof Abdo imad last updated on 14/Apr/18 $${find}\:\:{lim}_{{n}\rightarrow\infty} \:\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{x}^{\mathrm{2}} } \:{sin}^{{n}} {xdx}\: \\ $$ Commented by prof Abdo imad…
Question Number 33340 by prof Abdo imad last updated on 14/Apr/18 $${find}\:{a}\:{equivalent}\:{of} \\ $$$${A}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:\sqrt{\mathrm{1}\:+\left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} }\:\:{dx}\:\left({n}\rightarrow\infty\right) \\ $$ Commented by prof Abdo imad…