Question Number 133181 by mnjuly1970 last updated on 19/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……{nice}\:\:\:\:\:{calculus}… \\ $$$$\:\:\:\:\:\:{lim}\:_{{n}\rightarrow\infty} \left\{{n}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\frac{\mathrm{1}}{{n}+{k}}\right)^{\mathrm{2}} \right\}=?? \\ $$ Answered by Dwaipayan Shikari last updated on…
Question Number 2111 by 123456 last updated on 03/Nov/15 $$\mathrm{lets}\:{f}:\mathbb{R}^{\mathrm{2}} \rightarrow\mathbb{R} \\ $$$$\mathrm{does} \\ $$$$\frac{\partial{f}}{\partial{x}}=\frac{\partial}{\partial{x}}\underset{\mathrm{0}} {\overset{{x}} {\int}}\frac{\partial{f}}{\partial{y}}{dy}\:\:\:? \\ $$ Answered by prakash jain last updated…
Question Number 133180 by mathlove last updated on 19/Feb/21 Commented by mr W last updated on 19/Feb/21 $${no}\:{real}\:{solution}! \\ $$$${x}^{{x}} \geqslant\frac{\mathrm{1}}{\:\sqrt[{{e}}]{{e}}}\approx\mathrm{0}.\mathrm{692} \\ $$$$\frac{\mathrm{1}}{\mathrm{256}}<<\mathrm{0}.\mathrm{692} \\ $$$$\Rightarrow{x}^{{x}}…
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Question Number 133183 by bounhome last updated on 19/Feb/21 $$\int\sqrt{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }{dx}=…? \\ $$ Answered by physicstutes last updated on 19/Feb/21 $$\mathrm{let}\:{x}\:=\:\mathrm{sinh}\:\theta\:\Rightarrow\:{dx}\:=\:\mathrm{cosh}\:\theta\:{d}\theta \\ $$$$\mathrm{now},\:\int\sqrt{\left[\left(\mathrm{sinh}\:\theta\right)^{\mathrm{2}} +\mathrm{1}\right]^{\mathrm{3}}…
Question Number 133179 by abdullahquwatan last updated on 19/Feb/21 $$\mathrm{given}\:\mathrm{that}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{mx}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}\mathrm{x}+\mathrm{n}}{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{and}\:\mathrm{f}\:\mathrm{max}=\mathrm{7} \\ $$$$\mathrm{f}\:\mathrm{min}=−\mathrm{1}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{m}\:\mathrm{and}\:\mathrm{n} \\ $$ Commented by abdullahquwatan last updated on 20/Feb/21 $$\mathrm{thx} \\…
Question Number 133173 by john_santu last updated on 19/Feb/21 $$\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \:\sqrt{\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=? \\ $$ Answered by EDWIN88 last updated on 19/Feb/21 $$\mathrm{I}=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \sqrt{\mathrm{1}+\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}}…
Question Number 2103 by Yozzi last updated on 02/Nov/15 $${Find}\:{the}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{dx}}{{x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}}. \\ $$ Commented by Yozzi last updated on 02/Nov/15 $${Yup}. \\…
Question Number 133168 by bemath last updated on 19/Feb/21 Answered by floor(10²Eta[1]) last updated on 19/Feb/21 $$\mathrm{x}\rightarrow\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}} \\ $$$$\left(\mathrm{I}\right):\:\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{1}}\right)+\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}} \\ $$$$\mathrm{but}\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=\mathrm{x}−\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right),\:\mathrm{so} \\ $$$$\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{1}}\right)+\mathrm{x}−\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)=\frac{\mathrm{x}−\mathrm{1}}{\mathrm{x}} \\ $$$$\left(\mathrm{II}\right):\:\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{x}−\mathrm{1}}\right)−\mathrm{f}\left(\mathrm{1}−\mathrm{x}\right)=\frac{−\mathrm{x}^{\mathrm{2}}…
Question Number 67631 by Hassen_Timol last updated on 29/Aug/19 $$ \\ $$$$ \\ $$$$\:\:\:\mathrm{Can}\:\mathrm{you}\:\mathrm{please}\:\mathrm{tell}\:\mathrm{me},\:\mathrm{where}\:\mathrm{does}\:\mathrm{this}\: \\ $$$$\:\:\:\mathrm{formula}\:\mathrm{come}\:\mathrm{from}? \\ $$$$\:\:\:\mathrm{And}\:\mathrm{what}\:\mathrm{means}\:\mathrm{the}\:\mathrm{factorial}\:\mathrm{of}\:\mathrm{a}\:\mathrm{non}- \\ $$$$\:\:\:\mathrm{integer}\:\mathrm{number}? \\ $$$$ \\ $$$$\:\:\:\:\:\:\pi\:=\:\left(\frac{\mathrm{1}}{\mathrm{2}}!\right)^{\mathrm{2}} ×\:\mathrm{4}…