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Question Number 30356 Answers: 1 Comments: 0
$$\mathrm{If}\:\mathrm{sin}\:\mathrm{2}\theta=\:\mathrm{cos}\:\mathrm{3}\theta\:\:\mathrm{and}\:\theta\:\mathrm{is}\:\mathrm{an}\:\mathrm{acute} \\ $$$$\mathrm{angle},\:\mathrm{then}\:\mathrm{sin}\:\theta\:\mathrm{equals} \\ $$
Question Number 30355 Answers: 2 Comments: 0
Question Number 30354 Answers: 1 Comments: 0
$$\mathrm{If}\:\:{g}\left({x}\right)=\overset{{x}} {\int}_{\mathrm{0}} \mathrm{cos}^{\mathrm{4}} {t}\:{dt},\:\mathrm{then}\:{g}\:\left({x}+\pi\right)\:= \\ $$
Question Number 30350 Answers: 1 Comments: 0
Question Number 30331 Answers: 2 Comments: 0
Question Number 30340 Answers: 1 Comments: 0
Question Number 30321 Answers: 0 Comments: 3
$$\int_{−\infty} ^{\infty} \frac{\mathrm{e}^{\mathrm{a}{x}} }{\mathrm{e}^{{x}} +\mathrm{1}}{dx}=? \\ $$
Question Number 30323 Answers: 0 Comments: 0
Question Number 30299 Answers: 1 Comments: 5
Question Number 30282 Answers: 0 Comments: 3
$${Find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}cos}^{{n}} \:\left(\frac{\mathrm{2}\pi}{{n}}\right) \\ $$
Question Number 30348 Answers: 0 Comments: 1
Question Number 30267 Answers: 0 Comments: 7
$${Can}\:{We}\:{expand}\:{the}\:{following} \\ $$$${expression}? \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)\left(\mathrm{1}+\mathrm{3}{x}\right)......\left(\mathrm{1}+{nx}\right) \\ $$$${or}\:{is}\:{there}\:{any}\:{formula}\:{for}\:{this}? \\ $$
Question Number 30259 Answers: 1 Comments: 1
Question Number 30258 Answers: 2 Comments: 0
Question Number 30257 Answers: 1 Comments: 5
Question Number 30256 Answers: 0 Comments: 1
Question Number 30245 Answers: 1 Comments: 3
$$\mathrm{1}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} −\mathrm{7}^{\mathrm{2}} +\bullet\bullet\bullet+\mathrm{17}^{\mathrm{2}} −\mathrm{19}^{\mathrm{2}} \\ $$
Question Number 30244 Answers: 0 Comments: 0
$${if}\:{f}^{−\mathrm{1}} \:{exists}\:{and}\:{f}\:{is}\:{differentiable}\:{on}\:{R}\:,{the}\:{f}^{−\mathrm{1}} \:{is}\:{also}\:{differentiable}.\left({T}/{F}\right) \\ $$$$ \\ $$
Question Number 30235 Answers: 0 Comments: 3
$${find}\:{the}\:{sum}\:{of}\:{the}\:{infinite} \\ $$$${series}\: \\ $$$$\:\:\:\:\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2}}{{n}^{\mathrm{2}} }\right) \\ $$
Question Number 30233 Answers: 1 Comments: 1
Question Number 30221 Answers: 0 Comments: 0
Question Number 30220 Answers: 0 Comments: 1
$${let}\:{p}\left({x}\right)=\:{x}^{\mathrm{3}} \:{px}\:+{q} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{p}\:{have}\:{double}\:{roots}\Leftrightarrow\:\mathrm{4}{p}^{\mathrm{3}} \:+\mathrm{27}{q}^{\mathrm{2}} =\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\:{let}\:{suppose}\:{p}\:{have}\:\mathrm{3}\:{real}\:{roots}\:{differnts}\:{prove}\:{that} \\ $$$$\mathrm{4}{p}^{\mathrm{3}} \:+\mathrm{27}{q}^{\mathrm{2}} \:<\mathrm{0}. \\ $$
Question Number 30218 Answers: 0 Comments: 0
$${prove}\:{that}\:{D}\left({x}^{\mathrm{5}} −\mathrm{1},{x}^{\mathrm{2}} +{x}+\mathrm{1}\right)=\mathrm{1}. \\ $$
Question Number 30217 Answers: 0 Comments: 0
$${prove}\:{that}\:\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\frac{\mathrm{1}}{\sqrt{{n}}}\:−\frac{\mathrm{1}}{\sqrt{{n}+\mathrm{1}}}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)\sqrt{{n}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:{u}_{{n}} =\:\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{\mathrm{1}}{{k}\sqrt{{k}}}\:{is}\:{convergente}\:. \\ $$
Question Number 30216 Answers: 0 Comments: 0
$${let}\:{I}\left({x}\right)=\:\int_{\mathrm{0}} ^{\pi} \:\:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:{I}\left({x}\right)=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dt}}{{x}^{\mathrm{2}} \:+{cos}^{\mathrm{2}} {t}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:{I}\left({x}\right). \\ $$
Question Number 30215 Answers: 0 Comments: 0
$${let}\:{give}\:{J}\left({x}\right)=\:\frac{\mathrm{1}}{\pi}\:\int_{\mathrm{0}} ^{\pi} {cos}\left({xcost}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{J}^{'} \:{and}\:{J}^{''} \:{in}\:{form}\:{of}\:{integrals} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{J}^{'} \left({x}\right)=\frac{−{x}}{\pi}\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{2}} {t}\:{cos}\left({xcost}\right){dt}\:{and}\:{J}\:{is} \\ $$$${solution}\:{of}\:{d}.{e}.\:\:{xy}^{''} \:+{y}^{'} \:+{xy}=\mathrm{0} \\ $$
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