Question Number 34216 by abdo imad last updated on 02/May/18 $${let}\:{give}\:{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}\:{and}\:{J}\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{the}\:{existence}\:{of}\:{I}\:{and}\:{J} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{I}\:+{J}\:{and}\:\mathrm{2}{I}\:+{J} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{I}\:{and}\:{J}\:. \\ $$ Terms of…
Question Number 165273 by mnjuly1970 last updated on 28/Jan/22 $$ \\ $$$$\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\:\infty} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\:{n}\:.\:{e}^{\:\mathrm{1}−\:{x}^{\:\mathrm{2}} } }{\:\mathrm{1}\:+\:{n}^{\:\mathrm{2}} \:{x}^{\:\mathrm{2}} }\:{dx}\:=? \\ $$$$\:\:\:\:\:\:\:\:\:\:−−−−−− \\ $$ Answered by…
Question Number 165271 by mnjuly1970 last updated on 28/Jan/22 $$ \\ $$$$\:\:\:\:\:\mathrm{L}{et}\:,\:\:\:{f}\::\:\left[\:\mathrm{0}\:,\:\mathrm{1}\:\right]\:\rightarrow\:\mathbb{R}\:\:{is}\:{a}\:{continuous}\: \\ $$$$\:\:\:\:{function}\:,\:{prove}\:{that}\::\:\:\: \\ $$$$\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\:\infty} \:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{n}\:{f}\left({x}\right)}{\mathrm{1}+\:{n}^{\mathrm{2}} \:{x}^{\:\mathrm{2}} }\:{dx}\:=\:\frac{\pi}{\mathrm{2}}\:{f}\:\left(\mathrm{0}\:\right) \\ $$$$\:\:\:\:\:−−−\:{proof}\:−−− \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{S}_{\:{n}}…
Question Number 165255 by amin96 last updated on 28/Jan/22 $$\boldsymbol{\mathrm{nice}}\:\boldsymbol{\mathrm{integral}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{ln}}\left(\underset{\boldsymbol{\mathrm{m}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{m}}} \right)\boldsymbol{\mathrm{dx}}=? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$…
Question Number 99707 by student work last updated on 22/Jun/20 $$\int_{−\infty} ^{\infty} \mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \mathrm{dx}=? \\ $$ Commented by student work last updated on 22/Jun/20…
Question Number 99679 by Omer Alattas last updated on 22/Jun/20 Commented by maths mind last updated on 22/Jun/20 $${Quation}\:\:\mathrm{99485} \\ $$ Terms of Service Privacy…
Question Number 165218 by Lordose last updated on 27/Jan/22 $$\mathrm{Question}\:\mathrm{by}\:\boldsymbol{\mathrm{M}}.\boldsymbol{\mathrm{N}}\:\boldsymbol{\mathrm{July}} \\ $$$$\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{x}^{\mathrm{8}} \right)}{\mathrm{x}}\mathrm{dx} \\ $$$$\Phi\:\overset{\mathrm{x}=\mathrm{x}^{\frac{\mathrm{1}}{\mathrm{2}}} } {=}\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} \right)}{\mathrm{x}}\mathrm{dx} \\…
Question Number 34129 by abdo imad last updated on 01/May/18 $${find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}\right){ln}\left(\mathrm{1}+{x}\right){dx}\:. \\ $$ Commented by math khazana by abdo last updated on 02/May/18…
Question Number 34126 by abdo imad last updated on 30/Apr/18 $${let}\:{give}\:{n}\:{natural}\:{integr}\:{not}\:{o} \\ $$$${calculate}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{dx}}{\prod_{{k}=\mathrm{1}} ^{{n}} \left({x}^{\mathrm{2}} \:+{k}\right)}\:. \\ $$ Terms of Service Privacy…
Question Number 165183 by mathlove last updated on 27/Jan/22 Commented by Ar Brandon last updated on 27/Jan/22 $$\left({x}^{\mathrm{5}} +{x}^{\mathrm{2}} {y}^{\mathrm{3}} +{y}^{\mathrm{2}} {x}^{\mathrm{3}} +{y}^{\mathrm{5}} \right)=\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}}…