Question Number 153571 by mathdanisur last updated on 08/Sep/21 $$\mathrm{Find}\:\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}n}\centerdot\left(\frac{\pi^{\mathrm{2}} }{\mathrm{4}}\:-\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} ^{\mathrm{2}} \right)\:=\:? \\ $$$$\mathrm{where}\:\:\mathrm{a}_{\boldsymbol{\mathrm{n}}} =\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\mathrm{arctan}\left(\frac{\mathrm{1}}{\mathrm{k}^{\mathrm{2}} -\mathrm{k}+\mathrm{1}}\right) \\ $$ Answered by mnjuly1970…
Question Number 88032 by mathmax by abdo last updated on 07/Apr/20 $${decompose}\:{inside}\:{R}\left({x}\right)\:{the}\:{fraction} \\ $$$$\left.\mathrm{1}\right)\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{3}} \left({x}−\mathrm{2}\right)^{\mathrm{3}} } \\ $$$$\left.\mathrm{2}\right)\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$ Commented by jagoll…
Question Number 88033 by M±th+et£s last updated on 07/Apr/20 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{sin}\left({x}\right)}{{x}}{dx} \\ $$ Commented by mathmax by abdo last updated on 08/Apr/20 $${approximate}\:{value}\:{we}\:{have}\:{sinx}\:=\sum_{{n}=\mathrm{0}} ^{\infty}…
Question Number 153565 by talminator2856791 last updated on 08/Sep/21 $$\: \\ $$$$\:\mathrm{define}\:{d}\left({n}\right)\:\mathrm{to}\:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{digits}\:\: \\ $$$$\:\mathrm{of}\:{n}.\:\:\: \\ $$$$\:\mathrm{i}.\mathrm{e}\:\:{d}\left(\mathrm{1000}\right)\:=\:\mathrm{1}\:,\:\:\:\:{d}\left(\mathrm{999}\right)\:=\:\mathrm{27} \\ $$$$\: \\ $$$$\:\mathrm{find}\:\:{d}\left({d}\left({d}\left({d}\left({d}\left(\mathrm{5}^{\mathrm{10}^{\mathrm{100}} } \right)\right)\right)\right)\right) \\ $$$$\: \\…
Question Number 88029 by M±th+et£s last updated on 07/Apr/20 Answered by mind is power last updated on 08/Apr/20 $$−\mathrm{4}{sin}^{\mathrm{2}} \left(\mathrm{4}{x}\right)−\mathrm{8}{sin}\left(\mathrm{4}{x}\right)−\mathrm{9}{cos}^{\mathrm{2}} \left(\mathrm{4}{x}\right)+\mathrm{12}{cos}\left(\mathrm{4}{x}\right)−\mathrm{4}+\mathrm{6}{sin}\left(\mathrm{8}{x}\right) \\ $$$$=−\left(\mathrm{2}{sin}\left(\mathrm{4}{x}\right)−\mathrm{3}{cos}\left(\mathrm{4}{x}\right)+\mathrm{2}\right)^{\mathrm{2}} \\ $$$${f}\left({x}\right)=\frac{−\left(\mathrm{2}{sin}\left(\mathrm{4}{x}\right)−\mathrm{3}{cos}\left(\mathrm{4}{x}\right)+\mathrm{2}\right)^{\mathrm{2}}…
Question Number 22492 by Tinkutara last updated on 19/Oct/17 $$\mathrm{A}\:\mathrm{ball}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{400}\:\mathrm{g}\:\mathrm{travels}\:\mathrm{horizontally} \\ $$$$\mathrm{along}\:\mathrm{the}\:\mathrm{ground}\:\mathrm{and}\:\mathrm{collides}\:\mathrm{with}\:\mathrm{a} \\ $$$$\mathrm{wall}.\:\mathrm{The}\:\mathrm{velocity}-\mathrm{time}\:\mathrm{graph}\:\mathrm{below} \\ $$$$\mathrm{represents}\:\mathrm{the}\:\mathrm{motion}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{for}\:\mathrm{the} \\ $$$$\mathrm{first}\:\mathrm{1}.\mathrm{2}\:\mathrm{seconds}. \\ $$$$\mathrm{The}\:\mathrm{magnitude}\:\mathrm{of}\:\mathrm{average}\:\mathrm{force}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{ball}\:\mathrm{and}\:\mathrm{the}\:\mathrm{wall}\:\mathrm{is} \\ $$ Commented…
Question Number 88026 by Chi Mes Try last updated on 07/Apr/20 Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 22491 by Tinkutara last updated on 19/Oct/17 $$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{r}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\mathrm{1}\:−\:\mathrm{2}{x}\right)^{−\mathrm{1}/\mathrm{2}} \:\mathrm{is} \\ $$$$\left(\mathrm{1}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{2}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\mathrm{2}^{{r}} \:\left({r}!\right)^{\mathrm{2}} } \\ $$$$\left(\mathrm{3}\right)\:\frac{\left(\mathrm{2}{r}\right)!}{\left({r}!\right)^{\mathrm{2}} \:\mathrm{2}^{\mathrm{2}{r}} }…
Question Number 88027 by ajfour last updated on 07/Apr/20 Commented by ajfour last updated on 07/Apr/20 $${If}\:{released}\:{as}\:{shown},\:{end}\:{A} \\ $$$${of}\:{rod}\:{shall}\:{hit}\:{the}\:{smooth} \\ $$$${horizontal}\:{at}\:{what}\:{distance} \\ $$$${from}\:{corner}\:{C}.\:{Assume}\:{M}\:{is} \\ $$$${sufficiently}\:{greater}\:{than}\:{m}.…
Question Number 153563 by ZiYangLee last updated on 08/Sep/21 $$\int\:\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}\:}\:{dx}\:=? \\ $$ Answered by puissant last updated on 08/Sep/21 $${u}=\sqrt{\mathrm{2}{x}}\:\rightarrow\:{u}^{\mathrm{2}} =\mathrm{2}{x}\:\rightarrow\:{x}=\frac{{u}^{\mathrm{2}} }{\mathrm{2}}\:\Rightarrow\:{dx}={udu} \\ $$$${K}=\int\frac{\mathrm{1}}{\mathrm{1}+\sqrt{\mathrm{2}{x}}}{dx}=\int\frac{{u}}{\mathrm{1}+{u}}{du} \\…