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AllQuestion and Answers: Page 245

Question Number 196211 Answers: 1 Comments: 0

Show that cos((π/3)+i)=(1/4)(e+(1/e)) −((√3)/4)(e−(1/e))i

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{cos}\left(\frac{\pi}{\mathrm{3}}+\mathrm{i}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{e}+\frac{\mathrm{1}}{\mathrm{e}}\right)\:−\frac{\sqrt{\mathrm{3}}}{\mathrm{4}}\left(\mathrm{e}−\frac{\mathrm{1}}{\mathrm{e}}\right)\mathrm{i} \\ $$

Question Number 196209 Answers: 2 Comments: 0

Ω= ∫_0 ^( 1) (( (x−1)^( 2) )/(ln^2 (x))) dx= ? −−−−

$$ \\ $$$$\:\:\:\:\:\:\Omega=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\left({x}−\mathrm{1}\right)^{\:\mathrm{2}} }{\mathrm{ln}^{\mathrm{2}} \left({x}\right)}\:{dx}=\:? \\ $$$$\:\:\:\:\:−−−− \\ $$

Question Number 196205 Answers: 1 Comments: 0

What′s the value for !7 ?

$$\mathrm{What}'\mathrm{s}\:\mathrm{the}\:\mathrm{value}\:\mathrm{for}\:!\mathrm{7}\:? \\ $$

Question Number 196199 Answers: 1 Comments: 0

Solve: y′′′=((3y′(y′′)^2 )/(1+(y′)^2 ))

$$\mathrm{Solve}: \\ $$$${y}'''=\frac{\mathrm{3}{y}'\left({y}''\right)^{\mathrm{2}} }{\mathrm{1}+\left({y}'\right)^{\mathrm{2}} } \\ $$

Question Number 196198 Answers: 1 Comments: 0

a+b+c+d=4 prove that: (a/(a^3 +8))+(b/(b^3 +8))+(c/(c^3 +8))+(d/(d^3 +8))≤(4/9)

$${a}+{b}+{c}+{d}=\mathrm{4} \\ $$$${prove}\:{that}: \\ $$$$\frac{{a}}{{a}^{\mathrm{3}} +\mathrm{8}}+\frac{{b}}{{b}^{\mathrm{3}} +\mathrm{8}}+\frac{{c}}{{c}^{\mathrm{3}} +\mathrm{8}}+\frac{{d}}{{d}^{\mathrm{3}} +\mathrm{8}}\leq\frac{\mathrm{4}}{\mathrm{9}} \\ $$

Question Number 196185 Answers: 1 Comments: 0

x^(2a) = y^(2b) = z^(2c) ≠ 0 x^2 = yz ((ab+bc+ca)/(bc))

$$\:\: \mathrm{x}^{\mathrm{2a}} \:=\:\mathrm{y}^{\mathrm{2b}} \:=\:\mathrm{z}^{\mathrm{2c}} \:\neq\:\mathrm{0}\: \mathrm{x}^{\mathrm{2}} =\:\mathrm{yz} \\ $$$$\:\: \frac{\mathrm{ab}+\mathrm{bc}+\mathrm{ca}}{\mathrm{bc}} \\ $$

Question Number 196183 Answers: 2 Comments: 0

Question Number 196181 Answers: 0 Comments: 0

Question Number 196173 Answers: 1 Comments: 0

Calcul I_a =∫_a ^(1/a) ((lnx)/(1+x^2 ))dx (a>0)

$$\boldsymbol{{Calcul}}\:\boldsymbol{{I}}_{\boldsymbol{{a}}} =\int_{\boldsymbol{{a}}} ^{\frac{\mathrm{1}}{\boldsymbol{{a}}}} \frac{\boldsymbol{{lnx}}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}\:\left(\boldsymbol{{a}}>\mathrm{0}\right)\: \\ $$

Question Number 196172 Answers: 1 Comments: 2

Question Number 196166 Answers: 1 Comments: 1

for any a,b∈[−1,1] calculate: arccos(a)+arcsin(b)=? arcsin(a)+arcsin(b)=? arccos(a)+arccos(b)=?

$$\boldsymbol{{for}}\:\boldsymbol{{any}}\:\boldsymbol{{a}},\boldsymbol{{b}}\in\left[−\mathrm{1},\mathrm{1}\right]\:\:\boldsymbol{{calculate}}:\: \\ $$$$\boldsymbol{{arccos}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{arcsin}}\left(\boldsymbol{{b}}\right)=? \\ $$$$\boldsymbol{{arcsin}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{arcsin}}\left(\boldsymbol{{b}}\right)=? \\ $$$$\boldsymbol{{arccos}}\left(\boldsymbol{{a}}\right)+\boldsymbol{{arc}}{cos}\left(\boldsymbol{{b}}\right)=? \\ $$

Question Number 196164 Answers: 2 Comments: 0

calcul: lim_(n→+∞ ) Un=((((2n)!)/(n!n^n )))^(1/n)

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{calcul}: \\ $$$$ \\ $$$$\:\:\:{lim}_{{n}\rightarrow+\infty\:\:\:\:} {Un}=\sqrt[{{n}}]{\frac{\left(\mathrm{2}{n}\right)!}{{n}!{n}^{{n}} }} \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 196165 Answers: 3 Comments: 0

{: ((x^(x+y) =y^(24) )),((y^(x+y) =x^6 )) }⇒(x,y)=(?,?)

$$\left.\begin{matrix}{{x}^{{x}+{y}} ={y}^{\mathrm{24}} }\\{{y}^{{x}+{y}} ={x}^{\mathrm{6}} }\end{matrix}\right\}\Rightarrow\left({x},{y}\right)=\left(?,?\right) \\ $$

Question Number 196160 Answers: 0 Comments: 3

Question Number 196155 Answers: 1 Comments: 1

Question Number 196154 Answers: 2 Comments: 0

Question Number 196145 Answers: 1 Comments: 0

Calculer ∫^( +∞) _( (1/α)) e^(−αt^2 +2t) dt

$$\mathrm{Calculer}\:\underset{\:\frac{\mathrm{1}}{\alpha}} {\int}^{\:+\infty} {e}^{−\alpha{t}^{\mathrm{2}} +\mathrm{2}{t}} {dt} \\ $$

Question Number 196143 Answers: 2 Comments: 0

say you have 3 (different) books about mathematics, 4 (different) books about physics and 5 (different) books about chemistry. in how many ways can you arrange them in a shelf such that no two books from the same subject are adjacent?

$${say}\:{you}\:{have}\:\mathrm{3}\:\left({different}\right)\:{books} \\ $$$${about}\:{mathematics},\:\mathrm{4}\:\left({different}\right) \\ $$$${books}\:{about}\:{physics}\:{and}\:\mathrm{5}\:\left({different}\right) \\ $$$${books}\:{about}\:{chemistry}.\:{in}\:{how}\:{many} \\ $$$${ways}\:{can}\:{you}\:{arrange}\:{them}\:{in}\:{a}\:{shelf} \\ $$$${such}\:{that}\:{no}\:{two}\:{books}\:{from}\:{the}\:{same} \\ $$$${subject}\:{are}\:{adjacent}? \\ $$

Question Number 196135 Answers: 0 Comments: 0

Question Number 196132 Answers: 0 Comments: 0

Question Number 196121 Answers: 0 Comments: 0

In forest , a hunter obstains that Every morning a snake eats a mouse Every afternoon a scorpion kills a snake Every night a mouse corrodes a scorpion The 8^(th) day morning , there remains Only one of them , a mouse How many were they, in each species?

$${In}\:{forest}\:,\:{a}\:{hunter}\:{obstains}\:{that} \\ $$$${Every}\:{morning}\:{a}\:{snake}\:{eats}\:{a}\:{mouse} \\ $$$${Every}\:{afternoon}\:{a}\:{scorpion}\:{kills}\:{a}\:{snake} \\ $$$${Every}\:{night}\:{a}\:{mouse}\:{corrodes}\:{a}\:{scorpion} \\ $$$${The}\:\mathrm{8}^{{th}} \:{day}\:{morning}\:,\:{there}\:{remains} \\ $$$${Only}\:{one}\:{of}\:{them}\:,\:{a}\:{mouse} \\ $$$$ \\ $$$${How}\:{many}\:{were}\:{they},\:{in}\:{each}\:{species}? \\ $$

Question Number 196119 Answers: 5 Comments: 0

solve (√(100−x^2 ))+(√(64−x^2 ))=12

$${solve} \\ $$$$\sqrt{\mathrm{100}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{64}−{x}^{\mathrm{2}} }=\mathrm{12} \\ $$

Question Number 196117 Answers: 2 Comments: 0

Question Number 196115 Answers: 1 Comments: 0

Question Number 196111 Answers: 1 Comments: 0

Question Number 196107 Answers: 1 Comments: 0

help, please ! lim_(x → (𝛑/4)) ((cos((𝛑/4) −x)−tan x)/(1−sin((𝛑/4)+x))) = ???

$$ \\ $$$$\mathrm{help},\:\mathrm{please}\:! \\ $$$$\underset{{x}\:\rightarrow\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {{lim}}\:\frac{{cos}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}\:−{x}\right)−{tan}\:{x}}{\mathrm{1}−{sin}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}+{x}\right)}\:=\:???\: \\ $$

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