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Question Number 158354 Answers: 0 Comments: 2
$$\:\sqrt{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)}\:+\sqrt{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}\:=\sqrt[{\mathrm{4}}]{\mathrm{2cos}\:\mathrm{2}{x}} \\ $$
Question Number 158341 Answers: 1 Comments: 0
Question Number 158340 Answers: 2 Comments: 0
$$\left.\mathrm{1}\right)\:{Proven}\:{that}\:{by}\:{all}\:{n}\:\in\:{N}^{\ast} \\ $$$$\:\mathrm{2}!\mathrm{4}!..\left(\mathrm{2}{n}\right)!\geqslant\left(\left({n}+\mathrm{1}\right)!\right)^{{n}} \\ $$$$\left.\mathrm{2}\right)\:{Proven}\:{by}\:{recurring}\:{that}\: \\ $$$$\sum_{{p}=\mathrm{1}} ^{{n}} {pp}!=\left({n}+\mathrm{1}\right)!−\mathrm{1} \\ $$
Question Number 158334 Answers: 1 Comments: 3
$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{x}}\underset{{r}=\mathrm{1}} {\overset{{x}} {\sum}}{cos}\left(\frac{{r}\pi}{\mathrm{2}{x}}\right) \\ $$$${x}\in\mathbb{N} \\ $$
Question Number 158333 Answers: 0 Comments: 0
$$\int_{\left(\mathrm{1};\pi\right)} ^{\left(\mathrm{2};\pi\right)} \left(\mathrm{1}−\frac{\boldsymbol{\mathrm{y}}^{\mathrm{2}} }{\boldsymbol{\mathrm{x}}^{\mathrm{2}} }\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)\right)\boldsymbol{\mathrm{dx}}+\left(\boldsymbol{\mathrm{sin}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)+\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{cos}}\left(\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\right)\right)\boldsymbol{\mathrm{dy}}=? \\ $$$$\boldsymbol{\mathrm{OY}}\:\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{road}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{doesn}}'\boldsymbol{\mathrm{t}}\:\boldsymbol{\mathrm{cut}}\:\boldsymbol{\mathrm{your}}\:\boldsymbol{\mathrm{arrow}} \\ $$
Question Number 158335 Answers: 2 Comments: 0
$${if}\:\alpha,\beta,\gamma\:{are}\:{the}\:{angles}\:{of}\:{a}\:{triangle}, \\ $$$${find}\: \\ $$$$\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\alpha}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\beta}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\beta}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\gamma}}+\frac{\mathrm{1}}{\boldsymbol{\mathrm{tan}}\:\boldsymbol{\gamma}\:\boldsymbol{\mathrm{tan}}\:\boldsymbol{\alpha}}=? \\ $$
Question Number 158331 Answers: 0 Comments: 0
$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}}\:\left(\mathrm{n}\:+\:\mathrm{1}\right)}\:<\:\mathrm{2} \\ $$
Question Number 158330 Answers: 0 Comments: 1
Question Number 158327 Answers: 0 Comments: 0
$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:\:\:\:\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}\:\:\:+\:\:\:\mathrm{x}^{\mathrm{n}} }\:\:\:\:\mathrm{does}\:\mathrm{not}\:\mathrm{converge}\:\mathrm{uniformly}\:\mathrm{on}\:\:\left[\mathrm{0},\:\:\mathrm{2}\right] \\ $$$$\mathrm{by}\:\mathrm{showing}\:\mathrm{that}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{function}\:\mathrm{is}\:\mathrm{not}\:\mathrm{continuous}\:\mathrm{on}\:\:\:\:\left[\mathrm{0},\:\:\mathrm{2}\right] \\ $$
Question Number 158325 Answers: 2 Comments: 0
Question Number 158322 Answers: 0 Comments: 0
$$ \\ $$$$\:\:{prove}\:\:{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\:{tan}^{\:−\mathrm{1}} \:\left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}} }\:\right).{tan}^{\:−\mathrm{1}} \left(\:\frac{\mathrm{1}}{\mathrm{F}_{\:{n}+\mathrm{1}} }\:\right)=\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{8}} \\ $$$$\:\:\mathrm{F}{ibonacci}\:{numbers} \\ $$$$ \\ $$
Question Number 158320 Answers: 0 Comments: 0
$$ \\ $$$$\:\:\:{prove}\:\:{that}\:: \\ $$$$ \\ $$$$\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\mathrm{H}_{\:{n}} .\:\mathrm{F}_{{n}} }{\mathrm{2}^{\:{n}} }\:\:=\:{ln}\left(\mathrm{4}\right)\:+\:\frac{\mathrm{12}}{\:\sqrt{\mathrm{5}}}\:{ln}\left(\:\varphi\:\right) \\ $$$$\:\:\:\:\:\varphi\::\:\:\:\mathrm{Golden}\:\:\mathrm{ratio} \\ $$$$\:\:\:\:\:\:\mathrm{F}_{\:{n}} \::\:{fibonacci}\:{numbers} \\ $$$$ \\ $$
Question Number 158721 Answers: 1 Comments: 0
$$\mathrm{Find}: \\ $$$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{x}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:\mathrm{ax}\right)}\:\mathrm{dx}\:\:;\:\:\mathrm{a}>\mathrm{0} \\ $$$$ \\ $$
Question Number 158313 Answers: 1 Comments: 0
Question Number 158309 Answers: 3 Comments: 0
$${y}''+{y}'−\mathrm{2}{y}=−\mathrm{18}{te}^{−\mathrm{2}{t}} \\ $$
Question Number 158306 Answers: 0 Comments: 3
Question Number 158366 Answers: 1 Comments: 0
Question Number 158303 Answers: 1 Comments: 0
$$\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0} \\ $$$$\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\begin{cases}{\mathrm{8x}^{\mathrm{4}} \:+\:\mathrm{64y}^{\mathrm{4}} \:+\:\mathrm{216z}^{\mathrm{4}} \:+\:\mathrm{1728t}^{\mathrm{4}} \:=\:\mathrm{1}}\\{\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\:+\:\mathrm{t}\:=\:\mathrm{1}}\end{cases} \\ $$$$ \\ $$
Question Number 158425 Answers: 0 Comments: 0
Question Number 158424 Answers: 1 Comments: 0
Question Number 158301 Answers: 1 Comments: 1
$${proven}\:{that}\: \\ $$$$\mathrm{1}^{\mathrm{0}} =\mathrm{1}\:{et}\:{que}\:\mathrm{0}!=\mathrm{1} \\ $$
Question Number 158295 Answers: 0 Comments: 0
$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{sin}^{-\mathrm{1}} \:\mathrm{x}\:\mathrm{log}\left(\mathrm{1}\:+\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\:=\:? \\ $$$$ \\ $$
Question Number 158285 Answers: 0 Comments: 1
Question Number 158293 Answers: 3 Comments: 0
$$ \\ $$$$\:\:\:\:\:{question}# \\ $$$$\left.\mathrm{If}\:,\:\:\Omega\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\:\mathrm{2}} \left(\mathrm{1}−{x}^{\:\mathrm{4}} \right)}{{x}}\:{dx}=\:{a}\:\zeta\:{b}\right) \\ $$$$\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\:,\:\:\:\:\:{a}\:\:,\:{b}\:\:. \\ $$$$ \\ $$$$ \\ $$
Question Number 158317 Answers: 0 Comments: 0
Question Number 158290 Answers: 0 Comments: 0
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