Question Number 200417 by hardmath last updated on 18/Nov/23 $$\mathrm{find}:\:\:\:\Omega\:=\:\int_{\mathrm{1}} ^{\:\infty} \:\frac{\sqrt{\mathrm{x}}}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)}\:\mathrm{dx}\:=\:? \\ $$ Answered by witcher3 last updated on 18/Nov/23 $$\sqrt{\mathrm{x}}=\mathrm{y} \\ $$$$=\int_{\mathrm{1}}…
Question Number 200418 by mnjuly1970 last updated on 18/Nov/23 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{calculus}\:\:\left(\:\:\mathrm{I}\:\:\right)\:\: \\ $$$$\:\:\mathrm{I}{f}\:,\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\pi} \:\frac{\:{x}\:}{\mathrm{1}\:\:+\:\mathrm{sin}^{\mathrm{2}} \left({x}\right)}\:\mathrm{d}{x}\:=\:{a}\:\zeta\:\left(\:\mathrm{2}\:\right)\:\: \\ $$$$\:\:\:\:\:\:\:\Rightarrow\:\:\:\:{a}\:=\:?\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:{where}\:\:,\:\:\:\zeta\:\left({s}\:\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\:\mathrm{1}}{{n}^{\:{s}} } \\…
Question Number 200412 by hardmath last updated on 18/Nov/23 $$\mathrm{If} \\ $$$$\sqrt{\left(\mathrm{x}-\mathrm{6}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{1}\right)^{\mathrm{2}} }\:+\:\sqrt{\left(\mathrm{x}-\mathrm{9}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{find}:\:\:\:\mathrm{minumum}\:=\:? \\ $$ Answered by witcher3…
Question Number 200413 by Mingma last updated on 18/Nov/23 Commented by Mingma last updated on 18/Nov/23 Evaluate! Answered by witcher3 last updated on 18/Nov/23 $$\:\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}}…
Question Number 200415 by Mingma last updated on 18/Nov/23 Answered by witcher3 last updated on 18/Nov/23 $$\Leftrightarrow\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(−\frac{\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{x}−\mathrm{h}\right)}{\mathrm{x}−\left(\mathrm{x}−\mathrm{h}\right)}\right)=−\mathrm{f}'\left(\mathrm{x}\right) \\ $$$$\Leftrightarrow−\mathrm{f}''\left(\mathrm{x}\right)=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)\mathrm{x}^{\mathrm{n}} }{\mathrm{n}!} \\ $$$$=\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{0}\right)=−\mathrm{f}''\left(\mathrm{x}\right)…
Question Number 200403 by Anonim_X last updated on 18/Nov/23 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\left(\boldsymbol{{x}}^{\mathrm{2}} \:+\:\:\mathrm{1}\right)\boldsymbol{{dx}}}{\boldsymbol{{x}}\left(\boldsymbol{{x}}−\mathrm{1}\right)\left(\boldsymbol{{x}}+\mathrm{1}\right)}\:=\:?? \\ $$$$ \\ $$ Answered by cortano12 last updated on 18/Nov/23 $$\:\mathrm{I}\:=\:\int\:\frac{\mathrm{x}^{\mathrm{2}}…
Question Number 200395 by sonukgindia last updated on 18/Nov/23 Answered by witcher3 last updated on 18/Nov/23 $$\mathrm{I}_{\mathrm{10}} \rightarrow\mathrm{I}_{\mathrm{9}} \\ $$$$\mathrm{I}_{\mathrm{10}} =\frac{\mathrm{1}}{\mathrm{m}}\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{t}^{\frac{\mathrm{1}}{\mathrm{m}}−\mathrm{1}} }{\mathrm{1}+\mathrm{t}}\mathrm{dt} \\…
Question Number 200388 by mr W last updated on 18/Nov/23 $${find}\:{all}\:{values}\:{for}\:{k}\:{such}\:{that}\:{the}\:{eq}. \\ $$$${x}^{\mathrm{3}} −\mathrm{13}{x}+{k}=\mathrm{0}\:{has}\:{three}\:{integer}\:{roots}. \\ $$ Answered by Frix last updated on 18/Nov/23 $$\left({x}−{p}\right)\left({x}+\frac{{p}}{\mathrm{2}}−{q}\right)\left({x}+\frac{{p}}{\mathrm{2}}+{q}\right)=\mathrm{0} \\…
Question Number 200450 by hardmath last updated on 18/Nov/23 $${if}\:\:\:\frac{{sin}\mathrm{4}{x}}{{cos}\mathrm{6}{x}}\:=\:\mathrm{0}\:\:\:{find}\:\:\:{x}\:=\:? \\ $$ Answered by Frix last updated on 19/Nov/23 $$\mathrm{sin}\:\mathrm{4}{x}\:=\mathrm{0}\wedge\mathrm{cos}\:\mathrm{6}{x}\:\neq\mathrm{0}\:\Rightarrow\:{x}=\frac{{n}\pi}{\mathrm{2}} \\ $$ Terms of Service…
Question Number 200318 by sonukgindia last updated on 17/Nov/23 Answered by Sutrisno last updated on 17/Nov/23 $$=\int_{\mathrm{0}} ^{\pi} \frac{\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}}{\frac{\mathrm{1}}{{cos}^{\mathrm{2}} {x}}+\frac{{cos}^{\mathrm{2}} {x}}{{cos}^{\mathrm{2}} {x}}}{dx} \\ $$$$=\int_{\mathrm{0}}…