Question Number 15235 by arnabpapu550@gmail.com last updated on 08/Jun/17 $$\int\left(\mathrm{log}\:\sqrt{\mathrm{x}\:}\:\right)^{\mathrm{2}} \mathrm{dx}=? \\ $$ Answered by liday last updated on 18/Jun/17 $$\int\left({log}\sqrt{{x}}\right)^{\mathrm{2}} {dx}=\frac{\mathrm{1}}{\mathrm{4}}\int\left({logx}\right)^{\mathrm{2}} {dx}=\frac{\mathrm{1}}{\mathrm{4}}\left[{x}\left({logx}\right)^{\mathrm{2}} −\int{x}\centerdot\mathrm{2}{logx}\centerdot\frac{\mathrm{1}}{{x}}{dx}\right] \\…
Question Number 146307 by mnjuly1970 last updated on 12/Jul/21 $$ \\ $$$$\:\:\:\:\:\:\:\mathrm{S}_{{n}} \:=\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\:\mathrm{1}}{\left.{k}\:\left({k}+\mathrm{2}\right){k}+\mathrm{4}\right)} \\ $$$$\:\:\:\:\:\:\:\:{lim}_{\:{n}\rightarrow\infty} \:\left(\:\:\mathrm{S}_{\:{n}} \:\right)\:=\:? \\ $$ Answered by mathmax by…
Question Number 80764 by M±th+et£s last updated on 06/Feb/20 $${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\:{arctanh}\left({e}^{−\alpha{x}} \right){dx}=\frac{\mathrm{7}\zeta\left(\mathrm{3}\right)}{\mathrm{8}\alpha^{\mathrm{2}} } \\ $$ Answered by ~blr237~ last updated on 06/Feb/20…
Question Number 80752 by bagjamath last updated on 06/Feb/20 Commented by john santu last updated on 06/Feb/20 $${x}−\mathrm{2010}=\:{t}\: \\ $$$${x}−\mathrm{2009}={t}+\mathrm{1} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{0}} \left({t}^{\mathrm{2}} +{t}\right){dt}\:=\:\frac{\mathrm{1}}{\mathrm{3}}{t}^{\mathrm{3}}…
Question Number 146290 by SLVR last updated on 12/Jul/21 Commented by SLVR last updated on 12/Jul/21 $${Dear}\:{Olaf}\:{sir}\:{how}\:{r}\:{is}\:{taken}.. \\ $$$${how}\:{f}\left({x}\right)\:{was}\:{constructed}??? \\ $$$${kindly}\:{explain}… \\ $$$$ \\ $$…
Question Number 146212 by puissant last updated on 12/Jul/21 $$\mathrm{K}=\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }}\mathrm{dx} \\ $$ Answered by Ar Brandon last updated on 12/Jul/21 $$\mathrm{I}=\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }}=\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}+\left(\mathrm{x}^{\mathrm{3}/\mathrm{2}} \right)^{\mathrm{2}} }},\:\mathrm{u}=\mathrm{x}^{\mathrm{3}/\mathrm{2}}…
Question Number 146181 by mnjuly1970 last updated on 11/Jul/21 $$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}^{\:{n}} \:\left({n}+\mathrm{1}\:\right)\:\left(\:{n}\:+\:\mathrm{2}\:\right)}\:=? \\ $$ Answered by Olaf_Thorendsen last updated on 11/Jul/21 $$\mathrm{S}\:=\:\underset{{n}=\mathrm{0}}…
Question Number 15098 by tawa tawa last updated on 07/Jun/17 $$\int\:\mathrm{sin}^{\mathrm{8}} \left(\mathrm{x}\right)\:\mathrm{dx} \\ $$ Answered by mrW1 last updated on 07/Jun/17 $$=\int\left(\mathrm{1}−\mathrm{cos}^{\mathrm{2}} \:\mathrm{x}\right)\:\mathrm{sin}^{\mathrm{6}} \:\mathrm{x}\:\mathrm{dx} \\…
Question Number 146155 by mnjuly1970 last updated on 11/Jul/21 $$ \\ $$$$\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \:\left({Arcsin}\left({x}\right)\right)^{\:{n}} =\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\:\:\:\:\:\:{x}\:\in\:?\: \\ $$$$\:\:\:\:\:\:{Q}\::\:{mr}\:{liberty} \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$ Commented by mathmax…
Question Number 80612 by M±th+et£s last updated on 04/Feb/20 $$\:\Psi\left({x}\right)=\int_{\mathrm{1}} ^{{x}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{e}^{{t}} }}\:{dt}\:\:\:\:\:\forall{x}\in\mathbb{R} \\ $$$${prove}\:{that} \\ $$$$\Psi\left({x}\right)=\mathrm{2}{ln}\left(\frac{\mathrm{1}−\sqrt{\mathrm{1}−{e}^{{x}} }}{\mathrm{1}−\sqrt{\mathrm{1}−{e}}}\right)−{x}+\mathrm{1} \\ $$ Commented by mathmax by abdo…