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Question Number 161903 Answers: 2 Comments: 1

((1−sin(x)−cos(x))/(1+sin(x)−cos(x)))=???

$$\frac{\mathrm{1}−\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\mathrm{1}+\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}=??? \\ $$$$ \\ $$

Question Number 161900 Answers: 1 Comments: 0

0<x;y;z<1 (1-x)(1-y)(1-z)=xyz Find: Ω = min (((1-x)/(xy)) + ((1-y)/(yz)) + ((1-z)/(zx)))

$$\mathrm{0}<\mathrm{x};\mathrm{y};\mathrm{z}<\mathrm{1} \\ $$$$\left(\mathrm{1}-\mathrm{x}\right)\left(\mathrm{1}-\mathrm{y}\right)\left(\mathrm{1}-\mathrm{z}\right)=\mathrm{xyz} \\ $$$$\mathrm{Find}: \\ $$$$\Omega\:=\:\mathrm{min}\:\left(\frac{\mathrm{1}-\mathrm{x}}{\mathrm{xy}}\:+\:\frac{\mathrm{1}-\mathrm{y}}{\mathrm{yz}}\:+\:\frac{\mathrm{1}-\mathrm{z}}{\mathrm{zx}}\right) \\ $$

Question Number 161899 Answers: 2 Comments: 0

Find: 𝛀 =∫_( -∞) ^( ∞) (1/((1 + x^(2n) )^2 )) dx ; n∈Z

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\:-\infty} {\overset{\:\infty} {\int}}\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}\boldsymbol{\mathrm{n}}} \right)^{\mathrm{2}} }\:\mathrm{dx}\:\:;\:\:\mathrm{n}\in\mathbb{Z} \\ $$

Question Number 161884 Answers: 0 Comments: 2

Solve for real numbers: ((2^x + 2^(-1) ))^(1/7) = 1 + ((2^x - 2^(-1) ))^(1/7)

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt[{\mathrm{7}}]{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:+\:\mathrm{2}^{-\mathrm{1}} }\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{7}}]{\mathrm{2}^{\boldsymbol{\mathrm{x}}} \:-\:\mathrm{2}^{-\mathrm{1}} } \\ $$

Question Number 162231 Answers: 1 Comments: 0

nature et calcul ∫_0 ^1 ((lnx)/( (√(1−x))))dx

$${nature}\:{et}\:{calcul} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{lnx}}{\:\sqrt{\mathrm{1}−{x}}}{dx} \\ $$

Question Number 161875 Answers: 0 Comments: 0

Question Number 161888 Answers: 1 Comments: 5

help me ! solve this one : C_(40) ^(2n) = C_(40) ^(16+n)

$$\mathrm{help}\:\mathrm{me}\:! \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{one}\::\:\mathrm{C}_{\mathrm{40}} ^{\mathrm{2n}} \:=\:\mathrm{C}_{\mathrm{40}} ^{\mathrm{16}+\mathrm{n}} \\ $$

Question Number 161868 Answers: 1 Comments: 0

((√(2+(√3))) )^x + (1/(((√(2+(√3))))^x )) = 2^x x=?

$$\:\:\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\:\right)^{{x}} \:+\:\frac{\mathrm{1}}{\left(\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\right)^{{x}} }\:=\:\mathrm{2}^{{x}} \\ $$$$\:\:{x}=? \\ $$

Question Number 161867 Answers: 1 Comments: 0

lim_(x→∞) ((√(x^2 +8x+9)) −(√(x^2 +5x+4)) )^(4x) = ?

$$\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{{x}^{\mathrm{2}} +\mathrm{8}{x}+\mathrm{9}}\:−\sqrt{{x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{4}}\:\right)^{\mathrm{4}{x}} \:=\:? \\ $$

Question Number 161866 Answers: 1 Comments: 3

Question Number 161862 Answers: 0 Comments: 0

Question Number 161861 Answers: 1 Comments: 8

Prove that ((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!)) =n^2 +n−2

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\:\:\:\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={n}^{\mathrm{2}} +{n}−\mathrm{2} \\ $$

Question Number 161860 Answers: 1 Comments: 0

Simplify ((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!)) to n^2 +n−2

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Simplify} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{to} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{2} \\ $$

Question Number 161854 Answers: 1 Comments: 1

Question Number 161848 Answers: 0 Comments: 2

If tan (α )= 2 find the value of K=(( 1+sin( 8 α)−cos (8α ))/(1+sin( 8α ) + cos (8 α ))) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}{f}\:\:\:\:{tan}\:\left(\alpha\:\right)=\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{K}=\frac{\:\:\mathrm{1}+{sin}\left(\:\mathrm{8}\:\alpha\right)−{cos}\:\left(\mathrm{8}\alpha\:\right)}{\mathrm{1}+{sin}\left(\:\mathrm{8}\alpha\:\right)\:+\:{cos}\:\left(\mathrm{8}\:\alpha\:\right)}\:=? \\ $$$$ \\ $$$$ \\ $$

Question Number 161843 Answers: 0 Comments: 3

Q#161744 reposted with some change. Solve for integer numbers: (x/y) + (5/x) + ((y - 5)/5) = ((y + x)/(y + 5)) + ((5 + y)/(5 + x))

$${Q}#\mathrm{161744}\:{reposted}\:{with}\:{some}\:{change}. \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\boldsymbol{\mathrm{integer}}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\:+\:\frac{\mathrm{y}\:-\:\mathrm{5}}{\mathrm{5}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{x}}{\mathrm{y}\:+\:\mathrm{5}}\:+\:\frac{\mathrm{5}\:+\:\mathrm{y}}{\mathrm{5}\:+\:\mathrm{x}} \\ $$

Question Number 161839 Answers: 3 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +2x+2)^2 ))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

Question Number 161830 Answers: 1 Comments: 0

∫(dx/( (√(1−x^(14) ))))

$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{14}} }} \\ $$

Question Number 161823 Answers: 1 Comments: 0

Given that −1<x<1, find the expansion of ((3−2x)/((1+x)(4+x^2 ))) in ascending power of x, up to and including the term in x^3

$$\:{Given}\:{that}\:−\mathrm{1}<{x}<\mathrm{1},\:{find}\:{the} \\ $$$$\:{expansion}\:{of}\:\:\frac{\mathrm{3}−\mathrm{2}{x}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{4}+{x}^{\mathrm{2}} \right)}\:{in} \\ $$$$\:{ascending}\:{power}\:{of}\:{x},\:{up}\:{to}\:{and} \\ $$$$\:{including}\:{the}\:{term}\:{in}\:{x}^{\mathrm{3}} \\ $$

Question Number 161822 Answers: 1 Comments: 0

Find: Ω =lim_(n→∞) Σ_(k=1) ^n [Σ_(i=1) ^k (i + (1/4))]^( -1) = ?

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\left[\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\left(\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\right]^{\:-\mathrm{1}} =\:? \\ $$

Question Number 161820 Answers: 1 Comments: 0

Solve this differential equation: a ((∂L(α))/∂a) + b ((∂L(α))/∂b) = L(α) where: L(α) = ∫_( 0) ^( a) (√(a^2 sin^2 (t) + b^2 cos^2 (t))) dt

$$\mathrm{Solve}\:\mathrm{this}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$${a}\:\frac{\partial{L}\left(\alpha\right)}{\partial{a}}\:+\:{b}\:\frac{\partial{L}\left(\alpha\right)}{\partial{b}}\:=\:{L}\left(\alpha\right) \\ $$$$\mathrm{where}:\:{L}\left(\alpha\right)\:=\:\underset{\:\mathrm{0}} {\overset{\:\boldsymbol{{a}}} {\int}}\sqrt{{a}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({t}\right)\:+\:{b}^{\mathrm{2}} {cos}^{\mathrm{2}} \left({t}\right)}\:{dt} \\ $$

Question Number 161818 Answers: 1 Comments: 0

Show that: Φ =∫_( 0) ^( 1) (√((1 - x^2 )/(1 + x^2 ))) dx = ((√π)/4) (((Γ((1/4)))/(Γ((3/4)))) - 4 ((Γ((3/4)))/(Γ((1/4))))) where: Γ-Gamma function

$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\Phi\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\sqrt{\frac{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=\:\frac{\sqrt{\pi}}{\mathrm{4}}\:\left(\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}\:-\:\mathrm{4}\:\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\right) \\ $$$$\mathrm{where}:\:\Gamma-\mathrm{Gamma}\:\mathrm{function} \\ $$

Question Number 161817 Answers: 1 Comments: 0

Question Number 161815 Answers: 1 Comments: 0

Question Number 161811 Answers: 3 Comments: 0

Question Number 161800 Answers: 1 Comments: 2

((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!))=108 n=?

$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!}=\mathrm{108} \\ $$$${n}=? \\ $$

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