Question Number 135673 by liberty last updated on 14/Mar/21 $$\int\:\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}} +\mathrm{1}}\:{dx}\: \\ $$ Answered by Olaf last updated on 15/Mar/21 $$\Omega\:=\:\int\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}^{\mathrm{4}} +{x}^{\mathrm{2}}…
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Question Number 4575 by FilupSmith last updated on 08/Feb/16 $$\int_{{a}} ^{\:\infty} \frac{\mathrm{1}}{{x}}{dx}=\infty \\ $$$$ \\ $$$$\int_{{a}} ^{\:\infty} \lfloor\frac{\mathrm{1}}{{x}}\rfloor{dx}={S} \\ $$$${a}>\mathrm{0} \\ $$$$ \\ $$$$\mathrm{Is}\:{S}\:\mathrm{finite}?\:\mathrm{Can}\:\mathrm{you}\:\mathrm{solve}\:\mathrm{for}\:{S}? \\…
Question Number 135646 by metamorfose last updated on 14/Mar/21 $$\int\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right){ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)−{x}\:{dx}=…? \\ $$ Answered by Ñï= last updated on 15/Mar/21 $$\int\left[\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right){ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)−{x}\right]{dx} \\ $$$$=\int\left[\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right)\left({ln}\left({x}+\mathrm{1}\right)−{lnx}\right)−{x}\right]{dx} \\ $$$$=\int\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right){ln}\left({x}+\mathrm{1}\right){dx}−\int\left({x}+\frac{\mathrm{1}}{\mathrm{2}}\right){lnxdx}−\int{xdx} \\…
Question Number 135633 by metamorfose last updated on 14/Mar/21 $$\int\frac{\mathrm{1}}{{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{1}}{dx}=…? \\ $$ Answered by Ñï= last updated on 14/Mar/21 $$\int\frac{{dx}}{{x}^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{1}}\overset{{t}={x}^{\frac{\mathrm{1}}{\mathrm{3}}} } {=}\int\frac{\mathrm{3}{t}^{\mathrm{2}} {dt}}{{t}+\mathrm{1}}=\mathrm{3}\int\frac{\left({t}+\mathrm{1}\right)\left({t}−\mathrm{1}\right)+\mathrm{1}}{{t}+\mathrm{1}}{dt}=\mathrm{3}\int\left\{\left({t}−\mathrm{1}\right)+\frac{\mathrm{1}}{{t}+\mathrm{1}}\right\}{dt}…
Question Number 135627 by mnjuly1970 last updated on 14/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:……………..\:{calculus}\:… \\ $$$$\:\:\:\:\:\:\:{evaluation}:::::\:\:\:\boldsymbol{\phi}\overset{???} {=}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}\left({x}\right){ln}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:\:\:\:{solution}::::: \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}\overset{\langle{cos}\left({x}\right)={y}\rangle} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\mathrm{1}−{y}^{\mathrm{2}} \right){dy} \\ $$$$\:\:\:\:\:\:\:\:\:\:=−\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}}…
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Question Number 4535 by FilupSmith last updated on 05/Feb/16 $$\mathrm{Lets}\:\mathrm{say}\:\mathrm{we}\:\mathrm{have}\:\mathrm{three}\:\mathrm{points}: \\ $$$${A}\left(\mathrm{0},\:\mathrm{0}\right) \\ $$$${B}\left({x},\:{y}\right) \\ $$$${C}\left(\delta{x},\:\delta{y}\right) \\ $$$$ \\ $$$$\mathrm{Assuming}\:\mathrm{that}\:\mathrm{both}\:{B}\:\mathrm{and}\:{C}\:\mathrm{are}\:\mathrm{point} \\ $$$$\mathrm{on}\:\mathrm{a}\:\mathrm{fuction}\:{y}={f}\left({x}\right),\:\mathrm{we}\:\mathrm{can}\:\mathrm{calculate} \\ $$$$\mathrm{the}\:\mathrm{area}\:\mathrm{under}\:\mathrm{the}\:\mathrm{point}\:\mathrm{where}\:\mathrm{it}\:\mathrm{makes} \\…
Question Number 135610 by mnjuly1970 last updated on 14/Mar/21 $$\:\:\:\:\:\:\:\:\:\:….\:\mathscr{A}{dvanced}\:\:……\:\:\mathscr{C}{alculus}…. \\ $$$$\:\:\:\:\:\:\:\:\:{prove}\:{that}\:: \\ $$$$\:\:\:\begin{array}{|c|c|}{{i}\:::\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\mathrm{1}−\frac{{x}^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\right)\:={cos}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:\:\:\:\checkmark\:\:}\\{{ii}\:::\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\left(\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }\right)=\:{cosh}\left(\frac{\pi{x}}{\mathrm{2}}\right)\:\checkmark\checkmark}\\\hline\end{array}\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…………. \\…
Question Number 4521 by nabilah last updated on 05/Feb/16 $$\int\left(\mathrm{3}{x}+\mathrm{2}\hat {\right)}\mathrm{4} \\ $$ Answered by FilupSmith last updated on 05/Feb/16 $$\mathrm{If}\:\mathrm{you}\:\mathrm{mean}: \\ $$$$\int\left(\mathrm{3}{x}+\mathrm{2}\right)\mathrm{4}{dx} \\ $$$$=\mathrm{4}\int\left(\mathrm{3}{x}+\mathrm{2}\right){dx}…