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Question Number 157258    Answers: 0   Comments: 2

Solve for real numbers: (1/(sin^(2k) (x))) + (1/(cos^(2k) (x))) = 8 ; k∈Z

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}\boldsymbol{\mathrm{k}}} \left(\mathrm{x}\right)}\:+\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{2}\boldsymbol{\mathrm{k}}} \left(\mathrm{x}\right)}\:=\:\mathrm{8}\:\:\:;\:\:\:\mathrm{k}\in\mathbb{Z} \\ $$

Question Number 157257    Answers: 1   Comments: 0

∫ ((sin^6 x+cos^5 x)/(sin^2 x cos^2 x)) dx

$$\int\:\frac{\mathrm{sin}\:^{\mathrm{6}} {x}+\mathrm{cos}\:^{\mathrm{5}} {x}}{\mathrm{sin}\:^{\mathrm{2}} {x}\:\mathrm{cos}\:^{\mathrm{2}} {x}}\:{dx} \\ $$

Question Number 157251    Answers: 1   Comments: 0

# Nice Mathematics # ...calculation ... Ω :=∫_0 ^( 1) (( tanh^( −1) ((√( x)) ))/x) dx =^? (( π^( 2) )/4) −−−−−−−−−−−−− Ω :=^((√x) = t) 2∫_0 ^( 1) (( tanh^( −1) (t ))/t) dt :=^({tanh^( −1) (t )= (1/2) ln( ((1+t)/(1−t)) ) }) ∫_0 ^( 1) ((ln( 1+t )− ln(1−t ))/t) dt : = −Li_( 2) (−1 ) + Li_( 2) (1 ) :=^( {Li_( 2) (z )= Σ_(n=1) ^∞ (( z^( n) )/n^( 2) ) }) η (2) + ζ (2) := (π^( 2) /(12)) + (π^( 2) /6) = (( π^( 2) )/( 4)) ■ m.n

$$ \\ $$$$\:\:\:\:\:#\:\mathrm{Nice}\:\mathrm{Mathematics}\:# \\ $$$$\:\:\:\:\:\:\:...{calculation}\:... \\ $$$$\:\:\:\:\:\:\:\:\Omega\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left(\sqrt{\:{x}}\:\right)}{{x}}\:{dx}\:\overset{?} {=}\:\frac{\:\pi^{\:\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:\:−−−−−−−−−−−−− \\ $$$$\:\:\:\:\Omega\::\overset{\sqrt{{x}}\:=\:{t}} {=}\:\mathrm{2}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\:\:\:\:\:\::\overset{\left\{{tanh}^{\:−\mathrm{1}} \:\left({t}\:\right)=\:\frac{\mathrm{1}}{\mathrm{2}}\:{ln}\left(\:\frac{\mathrm{1}+{t}}{\mathrm{1}−{t}}\:\right)\:\right\}} {=}\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\:\mathrm{1}+{t}\:\right)−\:{ln}\left(\mathrm{1}−{t}\:\right)}{{t}}\:{dt} \\ $$$$\:\:\:\::\:\:=\:\:−\mathrm{Li}_{\:\mathrm{2}} \:\left(−\mathrm{1}\:\right)\:+\:\mathrm{Li}_{\:\mathrm{2}} \:\left(\mathrm{1}\:\right) \\ $$$$\:\:\:\:\::\overset{\:\left\{\mathrm{Li}_{\:\mathrm{2}} \:\left({z}\:\right)=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{z}^{\:{n}} }{{n}^{\:\mathrm{2}} }\:\right\}} {=}\:\:\eta\:\left(\mathrm{2}\right)\:+\:\zeta\:\left(\mathrm{2}\right)\: \\ $$$$\:\:\:\:\::=\:\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{12}}\:+\:\frac{\pi^{\:\mathrm{2}} }{\mathrm{6}}\:\:=\:\frac{\:\pi^{\:\mathrm{2}} }{\:\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:{m}.{n}\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 157247    Answers: 3   Comments: 0

F(x,y)=x^2 −2xy+6y^2 −12x+2y+45 find x &y such that F(x,y) minimum

$${F}\left({x},{y}\right)={x}^{\mathrm{2}} −\mathrm{2}{xy}+\mathrm{6}{y}^{\mathrm{2}} −\mathrm{12}{x}+\mathrm{2}{y}+\mathrm{45} \\ $$$${find}\:{x}\:\&{y}\:{such}\:{that}\:{F}\left({x},{y}\right)\:{minimum} \\ $$

Question Number 157241    Answers: 0   Comments: 0

Question Number 157231    Answers: 2   Comments: 0

SOLVE : ⌊ x ⌋ + ⌊2x ⌋ +⌊ 3x ⌋= 1 −−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:\mathcal{SOLVE}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\lfloor\:{x}\:\rfloor\:+\:\lfloor\mathrm{2}{x}\:\rfloor\:+\lfloor\:\mathrm{3}{x}\:\rfloor=\:\mathrm{1} \\ $$$$−−−−−−−−−− \\ $$$$ \\ $$

Question Number 157433    Answers: 0   Comments: 0

Question Number 157230    Answers: 1   Comments: 0

(3x+1)^(100) Find this max Koeffitcient

$$\left(\mathrm{3x}+\mathrm{1}\right)^{\mathrm{100}} \:\: \\ $$$$\mathrm{Find}\:\mathrm{this}\:\mathrm{max}\:\mathrm{Koeffitcient} \\ $$

Question Number 157438    Answers: 1   Comments: 0

find all subgroups of : a) grup (Z_6 , +) b) grup (Z_6 −{0}, ×)

$$\mathrm{find}\:\mathrm{all}\:\mathrm{subgroups}\:\mathrm{of}\:: \\ $$$$\left.\mathrm{a}\right)\:\mathrm{grup}\:\left(\mathrm{Z}_{\mathrm{6}} \:,\:+\right) \\ $$$$\left.\mathrm{b}\right)\:\mathrm{grup}\:\left(\mathrm{Z}_{\mathrm{6}} \:−\left\{\mathrm{0}\right\},\:×\right) \\ $$

Question Number 157436    Answers: 1   Comments: 0

if x;y;z≥0 then: 2^(x+y+z) + 2 ≥ 2^x + 2^y + 2^z

$$\mathrm{if}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\geqslant\mathrm{0}\:\:\mathrm{then}: \\ $$$$\mathrm{2}^{\boldsymbol{\mathrm{x}}+\boldsymbol{\mathrm{y}}+\boldsymbol{\mathrm{z}}} \:+\:\mathrm{2}\:\geqslant\:\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{y}}} \:+\:\mathrm{2}^{\boldsymbol{\mathrm{z}}} \\ $$$$ \\ $$

Question Number 157435    Answers: 1   Comments: 0

Question Number 157225    Answers: 0   Comments: 2

Question Number 157219    Answers: 0   Comments: 0

Σ_(0<n) (((−1)^(n−1) n)/(sinh(πn)))=(1/(4π)) prove

$$\underset{\mathrm{0}<\boldsymbol{\mathrm{n}}} {\sum}\frac{\left(−\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \boldsymbol{\mathrm{n}}}{\boldsymbol{\mathrm{sinh}}\left(\pi\boldsymbol{\mathrm{n}}\right)}=\frac{\mathrm{1}}{\mathrm{4}\pi}\:\:\:\:{prove} \\ $$

Question Number 157254    Answers: 1   Comments: 0

Show that ∫_0 ^1 (1/((x+1)(x+2)))dx = ln((4/3))

$$\mathrm{Show}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{1}}{\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)}{dx}\:=\:\mathrm{ln}\left(\frac{\mathrm{4}}{\mathrm{3}}\right) \\ $$

Question Number 157199    Answers: 1   Comments: 0

What is the general expression of the divergence(divV^⇢ )

$${What}\:{is}\:{the}\:{general}\:{expression}\:{of}\:{the}\:{divergence}\left({div}\overset{\dashrightarrow} {{V}}\right) \\ $$

Question Number 157227    Answers: 1   Comments: 0

x(3sin((√x))−2(√x))=sin^3 ((√x))

$$\boldsymbol{\mathrm{x}}\left(\mathrm{3}\boldsymbol{\mathrm{sin}}\left(\sqrt{\boldsymbol{\mathrm{x}}}\right)−\mathrm{2}\sqrt{\boldsymbol{\mathrm{x}}}\right)=\boldsymbol{\mathrm{sin}}^{\mathrm{3}} \left(\sqrt{\boldsymbol{\mathrm{x}}}\right) \\ $$

Question Number 157191    Answers: 1   Comments: 0

lim_(x→0) ((sin x − x + (1/6) x^3 )/x^5 ) = ? ( Without L′Hospital , Taylor or Maclaurin Series ) .

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}\:{x}\:−\:{x}\:+\:\frac{\mathrm{1}}{\mathrm{6}}\:{x}^{\mathrm{3}} }{{x}^{\mathrm{5}} }\:\:\:=\:\:? \\ $$$$\left(\:{Without}\:\:{L}'{Hospital}\:,\:{Taylor}\:\:{or}\:\:{Maclaurin}\:\:{Series}\:\right)\:. \\ $$

Question Number 157184    Answers: 1   Comments: 0

p(x)=x^3 −3ax^2 +(3a^2 +1)x−(a^3 +a) p(2)<0 prove that 2<a<3

$${p}\left({x}\right)={x}^{\mathrm{3}} −\mathrm{3}{ax}^{\mathrm{2}} +\left(\mathrm{3}{a}^{\mathrm{2}} +\mathrm{1}\right){x}−\left({a}^{\mathrm{3}} +{a}\right) \\ $$$${p}\left(\mathrm{2}\right)<\mathrm{0} \\ $$$${prove}\:{that}\:\mathrm{2}<{a}<\mathrm{3} \\ $$

Question Number 157220    Answers: 1   Comments: 0

x = (1/2) find (√(x + 1 + (√(x^2 + 1)))) = ?

$$\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{find}\:\:\sqrt{\mathrm{x}\:+\:\mathrm{1}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{1}}}\:=\:? \\ $$

Question Number 157177    Answers: 1   Comments: 0

Question Number 157217    Answers: 0   Comments: 2

show that ((cos2x+cos3x+cos8x)/(sin2x+sin3x+sin8x))=tanx

$${show}\:{that} \\ $$$$\:\frac{{cos}\mathrm{2}{x}+{cos}\mathrm{3}{x}+{cos}\mathrm{8}{x}}{{sin}\mathrm{2}{x}+{sin}\mathrm{3}{x}+{sin}\mathrm{8}{x}}={tanx} \\ $$

Question Number 157216    Answers: 1   Comments: 1

lim_(x→∞) (((n!)/n^n ))^(1/n) =? lim_(x→0) ((√(tan 4x))/(tan(√(4x))))=?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{n}!}{{n}^{{n}} }\right)^{\frac{\mathrm{1}}{{n}}} =? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{tan}\:\mathrm{4}{x}}}{\mathrm{tan}\sqrt{\mathrm{4}{x}}}=? \\ $$

Question Number 157222    Answers: 2   Comments: 0

Question Number 157223    Answers: 2   Comments: 0

Question Number 157175    Answers: 2   Comments: 1

Question Number 157171    Answers: 1   Comments: 0

calculate lim_(n→+∞) (((n+ln(n)+1)/((5+(√n))^2 )))

$${calculate}\:\underset{{n}\rightarrow+\infty} {{lim}}\left(\frac{{n}+{ln}\left({n}\right)+\mathrm{1}}{\left(\mathrm{5}+\sqrt{{n}}\right)^{\mathrm{2}} }\right) \\ $$

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