Question Number 195231 by cortano12 last updated on 28/Jul/23 $$\:\:\:\:\:\begin{array}{|c|}{\:\cancel{\underline{\underbrace{ }}}}\\\hline\end{array} \\ $$ Answered by MM42 last updated on 28/Jul/23 $${lim}_{{x}\rightarrow\mathrm{0}} \:\:\sqrt{\frac{\mathrm{1}−{cos}\sqrt{\pi{x}}}{{x}\left(\mathrm{1}+\sqrt{{cos}\sqrt{\pi{x}}}\right)}}\:\: \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}} \:\sqrt{\frac{\frac{\mathrm{1}}{\mathrm{2}}\pi{x}}{{x}\left(\mathrm{1}+\sqrt{\left.{cos}\sqrt{\pi{x}}\right)}\right.}}…
Question Number 195259 by mathlove last updated on 28/Jul/23 $${prove}\:{that} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{{e}^{{n}} \centerdot\left({n}!\right)}{{n}^{{n}} \:\sqrt{{n}}}=\sqrt{\mathrm{2}\pi} \\ $$ Commented by Frix last updated on 28/Jul/23 $$\mathrm{Easy}\:\mathrm{because}\:\mathrm{for}\:{n}\rightarrow\infty:\:{n}!\rightarrow\frac{{n}^{{n}}…
Question Number 195227 by York12 last updated on 28/Jul/23 $$ \\ $$$$\alpha_{\mathrm{1}} ^{\mathrm{3}} \left[\frac{\underset{{i}=\mathrm{2}} {\overset{{n}} {\prod}}\left({x}−\alpha_{{i}} \right)}{\underset{{i}=\mathrm{2}} {\overset{{n}} {\prod}}\left(\alpha_{\mathrm{1}} −\alpha_{{i}} \right)}\right]+\underset{{j}=\mathrm{2}} {\overset{{n}} {\sum}}\left(\alpha_{{j}} ^{\mathrm{3}} \left[\frac{\underset{{i}=\mathrm{1}}…
Question Number 195253 by Spillover last updated on 28/Jul/23 $${If}\:\mathrm{10sin}\:^{\mathrm{4}} {x}+\mathrm{15cos}\:^{\mathrm{4}} {x}=\mathrm{6}. \\ $$$${find}\:{the}\:{value}\:{of} \\ $$$$\mathrm{27cosec}\:^{\mathrm{6}} {x}+\mathrm{8sec}\:^{\mathrm{6}} {x} \\ $$$$ \\ $$ Commented by Frix…
Question Number 195252 by Spillover last updated on 28/Jul/23 $$\int_{\boldsymbol{{spillover}}} \:\:\:\:\:\:\frac{{dx}}{\:\sqrt{{e}^{\mathrm{5}{x}} }\:\sqrt{\left({e}^{\mathrm{2}{x}} +{e}^{−\mathrm{2}{x}} \right)^{\mathrm{3}} }} \\ $$ Answered by Frix last updated on 28/Jul/23 $$=\int\frac{\mathrm{e}^{\frac{{x}}{\mathrm{2}}}…
Question Number 195255 by Spillover last updated on 28/Jul/23 $$\int\frac{{dx}}{\mathrm{cos}\:^{\mathrm{3}} {x}\sqrt{\mathrm{4sin}\:{x}\mathrm{cos}\:{x}}} \\ $$ Answered by Frix last updated on 28/Jul/23 $$\left[\mathrm{Using}\:{t}=\sqrt{\mathrm{tan}\:{x}}\right] \\ $$$$=\int\left({t}^{\mathrm{4}} +\mathrm{1}\right){dt}=…=\frac{\mathrm{5}+\mathrm{tan}^{\mathrm{2}} \:{x}}{\mathrm{5}}\sqrt{\mathrm{tan}\:{x}}\:+{C}…
Question Number 195254 by Spillover last updated on 02/Aug/23 $$\int^{\boldsymbol{{spillover}}} \frac{\mathrm{sin}\:^{\mathrm{2}} {x}\mathrm{cos}\:^{\mathrm{2}} {x}}{\left(\mathrm{sin}\:^{\mathrm{5}} {x}+\mathrm{cos}\:^{\mathrm{3}} {x}\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{sin}\:^{\mathrm{3}} {x}\mathrm{cos}\:^{\mathrm{2}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}\right)^{\mathrm{2}} }{dx} \\ $$ Answered by Spillover…
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Question Number 195248 by arkanshh last updated on 28/Jul/23 Commented by Frix last updated on 28/Jul/23 $$\mathrm{Approximate} \\ $$$${x}_{\mathrm{1}} \approx−.\mathrm{811328} \\ $$$${x}_{\mathrm{2}} \approx\mathrm{4}.\mathrm{53236} \\ $$…
Question Number 195251 by Spillover last updated on 28/Jul/23 $${If}\:\:{x}^{\left[\mathrm{16}\left(\mathrm{log}\:_{\mathrm{5}} {x}\right)^{\mathrm{3}} −\mathrm{68log}\:_{\mathrm{5}} {x}\right]} =\mathrm{5}^{−\mathrm{16}} \: \\ $$$$\:{then}\:{Find}\:{the}\:{the}\:{product}\:{of}\:{x} \\ $$$$ \\ $$ Commented by Frix last…