Question Number 86611 by Ar Brandon last updated on 30/Mar/20 $$\int_{\mathrm{1}} ^{{e}} \frac{\mathrm{ln}\:\mathrm{x}}{\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$ Answered by Ar Brandon last updated on 30/Mar/20 Terms of…
Question Number 152147 by john_santu last updated on 26/Aug/21 Answered by Ar Brandon last updated on 26/Aug/21 $${I}_{{n}} =\int_{−\mathrm{1}} ^{\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{n}}}−\mid{x}\mid\right){dx} \\ $$$$\:\:\:\:=\mathrm{2}\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 152141 by Ar Brandon last updated on 26/Aug/21 $$\int_{\mathrm{0}} ^{+\infty} \frac{\left(\mathrm{sin}{x}\right)^{\mathrm{2}{n}+\mathrm{1}} }{{x}}{dx}=\frac{\pi}{\mathrm{2}^{\mathrm{2}{n}+\mathrm{1}} }\begin{pmatrix}{\mathrm{2}{n}}\\{{n}}\end{pmatrix} \\ $$ Answered by Olaf_Thorendsen last updated on 26/Aug/21 $$\mathrm{I}_{{n}}…
Question Number 21071 by Tinkutara last updated on 12/Sep/17 $$\mathrm{The}\:\mathrm{values}\:\mathrm{of}\:'{k}'\:\mathrm{for}\:\mathrm{which}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mid{x}\mid^{\mathrm{2}} \left(\mid{x}\mid^{\mathrm{2}} \:−\:\mathrm{2}{k}\:+\:\mathrm{1}\right)\:=\:\mathrm{1}\:−\:{k}^{\mathrm{2}} ,\:\mathrm{has} \\ $$$$\mathrm{repeated}\:\mathrm{roots},\:\mathrm{when}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\left(\mathrm{1}\right)\:\left\{\mathrm{1},\:−\mathrm{1}\right\} \\ $$$$\left(\mathrm{2}\right)\:\left\{\mathrm{0},\:\mathrm{1}\right\} \\ $$$$\left(\mathrm{3}\right)\:\left\{\mathrm{0},\:−\mathrm{1}\right\} \\ $$$$\left(\mathrm{4}\right)\:\left\{\mathrm{2},\:\mathrm{3}\right\}…
Question Number 152140 by Ar Brandon last updated on 26/Aug/21 $${I}=\int_{\mathrm{0}} ^{\mathrm{2n}\pi} \mathrm{max}\left(\mathrm{sin}{x},\:\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}{x}\right)\right){dx} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 21070 by Tinkutara last updated on 11/Sep/17 $$\mathrm{Let}\:\mathrm{us}\:\mathrm{consider}\:\mathrm{an}\:\mathrm{equation}\:{f}\left({x}\right)\:=\:{x}^{\mathrm{3}} \\ $$$$−\:\mathrm{3}{x}\:+\:{k}\:=\:\mathrm{0}.\:\mathrm{Then}\:\mathrm{the}\:\mathrm{values}\:\mathrm{of}\:{k}\:\mathrm{for} \\ $$$$\mathrm{which}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{has} \\ $$$$\mathrm{1}.\:\mathrm{Exactly}\:\mathrm{one}\:\mathrm{root}\:\mathrm{which}\:\mathrm{is}\:\mathrm{positive}, \\ $$$$\mathrm{then}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\mathrm{2}.\:\mathrm{Exactly}\:\mathrm{one}\:\mathrm{root}\:\mathrm{which}\:\mathrm{is}\:\mathrm{negative}, \\ $$$$\mathrm{then}\:{k}\:\mathrm{belongs}\:\mathrm{to} \\ $$$$\mathrm{3}.\:\mathrm{One}\:\mathrm{negative}\:\mathrm{and}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{root} \\…
Question Number 152142 by Ar Brandon last updated on 26/Aug/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}}{\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{ln}\left(\mathrm{ln}\frac{\mathrm{1}}{{x}}\right){dx}=−\frac{\gamma}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{ln}\frac{\mathrm{6}\sqrt{\mathrm{3}}}{\pi}+\frac{\pi\sqrt{\mathrm{3}}}{\mathrm{27}}\left(\mathrm{5ln2}\pi−\mathrm{6ln}\Gamma\left(\frac{\mathrm{1}}{\mathrm{6}}\right)\right) \\ $$ Answered by mindispower last updated on 28/Aug/21 $$\int_{\mathrm{0}}…
Question Number 86602 by rizababa last updated on 29/Mar/20 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 21067 by youssoufab last updated on 11/Sep/17 $$\forall{n}\in\mathbb{N},\:{prove}\:\mathrm{9}\mid\left[{n}^{\mathrm{3}} +\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\left({n}+\mathrm{2}\right)^{\mathrm{3}} \right] \\ $$ Answered by dioph last updated on 12/Sep/17 $${n}^{\mathrm{3}} +\left({n}+\mathrm{1}\right)^{\mathrm{3}} +\left({n}+\mathrm{2}\right)^{\mathrm{3}}…
Question Number 86603 by Ar Brandon last updated on 29/Mar/20 $$\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\frac{{e}^{{x}} }{\left(\mathrm{1}+{x}\right)^{{n}} } \\ $$ Commented by abdomathmax last updated on 29/Mar/20 $${let}\:{f}\left({x}\right)=\frac{{e}^{{x}} }{\left(\mathrm{1}+{x}\right)^{{n}}…