Question Number 108738 by mnjuly1970 last updated on 19/Aug/20 $$\:\:\:\:\:\:\:\:{please}:\:\:\:\:\:^{\ast} \mathrm{prove}^{\ast} :::: \\ $$$$\:\:\:\:\:\mathrm{1}.^{\mathrm{important}} \:\:\:\:\mathrm{lim}_{\mathrm{z}\rightarrow\mathrm{1}} \left(\zeta\:\left(\mathrm{z}\right)\:−\frac{\mathrm{1}}{\mathrm{z}−\mathrm{1}}\:\right)=\:\gamma\:\:\:\left(\mathrm{euler}\:\mathrm{constant}\right) \\ $$$$\:\:\:\:\mathrm{2}.\:\overset{\mathrm{important}} {\:}\:\:\int_{\mathrm{0}} ^{\:\infty} \left(\mathrm{cos}\left(\mathrm{x}\right)−\frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\right)\frac{\mathrm{dx}}{\mathrm{x}}\:=−\:\gamma \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..\mathscr{M}.\mathscr{N}….. \\…
Question Number 43191 by MASANJA J last updated on 08/Sep/18 $${integrate}\:{by}\:{use}\:{a}\:{partial}\:{friction} \\ $$$$\int\frac{{lnx}}{\left(\mathrm{1}+{x}\right)^{\mathrm{2}} } \\ $$ Commented by mondodotto@gmail.com last updated on 09/Sep/18 $$\mathrm{did}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{by}\:\mathrm{parts}? \\…
Question Number 43190 by MASANJA J last updated on 08/Sep/18 $${a}\:{point}\:{move}\:{in}\:{such}\:{away}\:{that}\:{its}\: \\ $$$${its}\:{distance}\:{from}\:{the}\:{x}−{axis}\:{is}\:{alwa} \\ $$$${yas}\frac{\mathrm{1}}{\mathrm{5}}\:{its}\:{distance}\:{from}\:{origin}. \\ $$$${find}\:{the}\:{equetion}\:{of}\:{its}\:{path}. \\ $$ Commented by MrW3 last updated on…
Question Number 108723 by 150505R last updated on 18/Aug/20 Commented by bemath last updated on 19/Aug/20 $${I}=\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\mathrm{cos}\:\:\left({x}−\frac{\pi}{\mathrm{4}}\right)\right){dx} \\ $$$$\:=\:\left[\:{x}\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\right)\:\right]_{\mathrm{0}} ^{\pi/\mathrm{2}} +\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\mathrm{ln}\:\left(\mathrm{cos}\:\:\left({x}−\frac{\pi}{\mathrm{4}}\right)\right){dx}…
Question Number 108710 by mathmax by abdo last updated on 18/Aug/20 $$\mathrm{calculate}\:\:\int_{−\infty} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{\mathrm{x}^{\mathrm{2}} } }{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$ Commented by sunilbaraskar last updated…
Question Number 43170 by maxmathsup by imad last updated on 07/Sep/18 $${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\left[{n}\:{e}^{−{x}} \right]{dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} . \\ $$ Commented…
Question Number 108697 by 150505R last updated on 18/Aug/20 Answered by mathmax by abdo last updated on 18/Aug/20 $$\mathrm{I}\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(\mathrm{lnx}\right)^{\mathrm{2}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:\Rightarrow\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\left(\mathrm{lnx}\right)^{\mathrm{2}}…
Question Number 43158 by MASANJA J last updated on 07/Sep/18 $$\int{cosecxdx} \\ $$ Commented by maxmathsup by imad last updated on 07/Sep/18 $${let}\:{I}\:=\:\int\:\:\:\frac{{dx}}{{sin}\left({x}\right)}\:{changement}\:{tan}\left(\frac{{x}}{\mathrm{2}}\right)={t}\:{give} \\ $$$${I}\:=\:\int\:\:\:\:\frac{\mathrm{1}}{\frac{\mathrm{2}{t}}{\mathrm{1}+{t}^{\mathrm{2}}…
Question Number 43159 by MASANJA J last updated on 07/Sep/18 Answered by alex041103 last updated on 08/Sep/18 $${For}\:\int\frac{{dx}}{\:\sqrt{{x}+\mathrm{15}}}\:: \\ $$$$\int\frac{{dx}}{\:\sqrt{{x}+\mathrm{15}}}\:=\:\int\frac{{d}\left({x}+\mathrm{15}\right)}{\:\sqrt{{x}+\mathrm{15}}}=\int{u}^{−\mathrm{1}/\mathrm{2}} {du}= \\ $$$$=\mathrm{2}{u}^{\mathrm{1}/\mathrm{2}} +{C}=\mathrm{2}\sqrt{{x}+\mathrm{15}}\:+{C} \\…
Question Number 108692 by 150505R last updated on 18/Aug/20 Answered by Dwaipayan Shikari last updated on 18/Aug/20 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{2}{log}\left({tan}\theta\right)}{{tan}^{\mathrm{2}} \theta+\mathrm{1}}{sec}^{\mathrm{2}} \theta{d}\theta\:\:\:\left({x}={tan}\theta,\:\mathrm{1}={sec}^{\mathrm{2}} \theta\frac{{d}\theta}{{dx}}\right) \\ $$$$\mathrm{2}\int_{\mathrm{0}}…