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 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 $$\left[\mathrm{Using}\:{t}=\sqrt{\mathrm{tan}\:{x}}\right] \\ $$$$=\int\left({t}^{\mathrm{4}} +\mathrm{1}\right){dt}=…=\frac{\mathrm{5}+\mathrm{tan}^{\mathrm{2}} \:{x}}{\mathrm{5}}\sqrt{\mathrm{tan}\:{x}}\:+{C}…
Question Number 195252 by Spillover last updated on 28/Jul/23 $$\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 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…
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…
Question Number 195194 by Abdullahrussell last updated on 26/Jul/23 Answered by Frix last updated on 26/Jul/23 $${a}\:\:\:\:\:{b}\:\:\:\:\:{c}\:\:\:\:\:{d} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{3}\:\:\:\:\mathrm{15}\:\:\:\mathrm{10} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{4}\:\:\:\:\mathrm{12}\:\:\:\:\mathrm{6} \\ $$$$\mathrm{2}\:\:\:\:\:\mathrm{6}\:\:\:\:\mathrm{12}\:\:\:\:\mathrm{4} \\ $$$$\mathrm{2}\:\:\:\:\mathrm{10}\:\:\:\mathrm{15}\:\:\:\:\mathrm{3}…
Question Number 195124 by Shlock last updated on 25/Jul/23 Answered by witcher3 last updated on 25/Jul/23 $$\sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Leftrightarrow\mathrm{x}+\mathrm{y}+\mathrm{2}\sqrt{\mathrm{xy}}\leqslant\mathrm{xy}+\mathrm{x}+\mathrm{y}+\mathrm{1}\Leftrightarrow\mathrm{xy}+\mathrm{1}\geqslant\mathrm{2}\sqrt{\mathrm{xy}},\mathrm{AM}−\mathrm{GM} \\ $$$$\Rightarrow\forall\left(\mathrm{x},\mathrm{y}\right)\in\mathbb{R}_{+} \sqrt{\mathrm{x}}+\sqrt{\mathrm{y}}\leqslant\sqrt{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{y}+\mathrm{1}\right)} \\ $$$$\Rightarrow\forall\left(\mathrm{a},\mathrm{b}\right)\in\left[\mathrm{1},\infty\left[^{\mathrm{2}} \right.\right.…
Question Number 195121 by mathlove last updated on 25/Jul/23 $${x}^{\mathrm{2}} −{x}−\mathrm{1}=\mathrm{0} \\ $$$${x}^{\mathrm{8}} +\mathrm{2}{x}^{\mathrm{7}} −\mathrm{47}{x}=? \\ $$ Answered by qaz last updated on 25/Jul/23 $${x}^{\mathrm{8}}…