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Question Number 221233 Answers: 1 Comments: 2

if a,b,c,d,e,f > 0 and abcdef = 1 , then (1/( (√(1 + ad)))) + (1/( (√(1 + be)))) + (1/( (√(1 + cf)))) ≤ (3/( (√2))) Profosed by Craciun Georghe

$$ \\ $$$$\:\mathrm{if}\:{a},{b},{c},{d},{e},{f}\:>\:\mathrm{0}\:\mathrm{and}\:{abcdef}\:=\:\mathrm{1}\:, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{then} \\ $$$$\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{ad}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{be}}}\:+\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}\:+\:{cf}}}\:\leqslant\:\frac{\mathrm{3}}{\:\sqrt{\mathrm{2}}}\: \\ $$$$\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Profosed}\:\mathrm{by}\:\mathrm{Craciun}\:\mathrm{Georghe} \\ $$

Question Number 221228 Answers: 1 Comments: 0

Σ_(n = 1) ^∞ Σ_(m = 1) ^∞ (1/((m^2 + n^2 )^2 ))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\underset{{m}\:=\:\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\left({m}^{\mathrm{2}} \:+\:{n}^{\mathrm{2}} \right)^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 221218 Answers: 0 Comments: 1

Question Number 221203 Answers: 1 Comments: 0

∫ tan(((1/x)/(sec(x)))) + ((1 − sec(x))/(sec(x))) dx

$$ \\ $$$$\:\:\:\:\:\int\:\mathrm{tan}\left(\frac{\frac{\mathrm{1}}{{x}}}{\mathrm{sec}\left({x}\right)}\right)\:+\:\frac{\mathrm{1}\:−\:\mathrm{sec}\left({x}\right)}{\mathrm{sec}\left({x}\right)}\:\mathrm{d}{x} \\ $$$$ \\ $$

Question Number 221196 Answers: 1 Comments: 0

Question Number 221195 Answers: 1 Comments: 1

Question Number 221188 Answers: 1 Comments: 2

Question Number 221185 Answers: 5 Comments: 0

Question Number 221184 Answers: 3 Comments: 2

Question Number 221179 Answers: 0 Comments: 0

Question Number 221170 Answers: 3 Comments: 1

Question Number 221180 Answers: 0 Comments: 0

Question Number 221168 Answers: 1 Comments: 0

Question Number 221154 Answers: 0 Comments: 1

f(x)= (x/(∣ x ∣ + 1)) f(f(f(f(x)))) =?

$$\:{f}\left({x}\right)=\:\frac{{x}}{\mid\:{x}\:\mid\:+\:\mathrm{1}} \\ $$$$\:\:{f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\:=? \\ $$

Question Number 221153 Answers: 1 Comments: 1

f(x)= (1/2^x ) + (1/3^x ) + (1/4^x ) + ... +(1/(4000^x )) f(2) + f(3) + f(4)+ ... =?

$$\:\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{3}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{4}^{{x}} }\:+\:...\:+\frac{\mathrm{1}}{\mathrm{4000}^{{x}} } \\ $$$$\:\:{f}\left(\mathrm{2}\right)\:+\:{f}\left(\mathrm{3}\right)\:+\:{f}\left(\mathrm{4}\right)+\:...\:=? \\ $$

Question Number 221151 Answers: 0 Comments: 1

((√(x^2 −x−(√(x^2 −x−(√(x^2 −x−(√(...))))))))/( ((x^2 (√(x ((x^2 (√(x...))))^(1/3) ))))^(1/3) )) = (3/4) ⇒ (2/x) =?

$$\:\frac{\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{{x}^{\mathrm{2}} −{x}−\sqrt{...}}}}}{\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} \:\sqrt{{x}\:\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} \:\sqrt{{x}...}}}}}\:=\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$$$\:\Rightarrow\:\frac{\mathrm{2}}{{x}}\:=?\: \\ $$

Question Number 221135 Answers: 1 Comments: 2

South Korean Grade 12 math Prove log_a M^n =nlog_a M Using below: When M=a^x , log_a M=x When N=a^y , log_a N=y MN=a^x ×a^y =a^(x+y) So, log_a (MN)=log_a (a^(x+y) )=x+y=log_a M+log_a N

$$\mathrm{South}\:\mathrm{Korean}\:\mathrm{Grade}\:\mathrm{12}\:\mathrm{math} \\ $$$$\mathrm{Prove}\:\mathrm{log}_{{a}} {M}^{{n}} ={n}\mathrm{log}_{{a}} {M} \\ $$$$\mathrm{Using}\:\mathrm{below}: \\ $$$$\mathrm{When}\:{M}={a}^{{x}} ,\:\mathrm{log}_{{a}} {M}={x} \\ $$$$\mathrm{When}\:{N}={a}^{{y}} ,\:\mathrm{log}_{{a}} {N}={y} \\ $$$${MN}={a}^{{x}} ×{a}^{{y}} ={a}^{{x}+{y}} \\ $$$$\mathrm{So},\:\mathrm{log}_{{a}} \left({MN}\right)=\mathrm{log}_{{a}} \left({a}^{{x}+{y}} \right)={x}+{y}=\mathrm{log}_{{a}} {M}+\mathrm{log}_{{a}} {N} \\ $$

Question Number 221129 Answers: 1 Comments: 0

prove Contour integral repreasentation ((p),(q) )=(1/(2πi)) ∮_( C) (1−z)^p z^(−q) (dz/z)

$$\mathrm{prove} \\ $$$$\mathrm{Contour}\:\mathrm{integral}\:\mathrm{repreasentation} \\ $$$$\begin{pmatrix}{{p}}\\{{q}}\end{pmatrix}=\frac{\mathrm{1}}{\mathrm{2}\pi\boldsymbol{{i}}}\:\oint_{\:{C}} \:\left(\mathrm{1}−{z}\right)^{{p}} {z}^{−{q}} \:\frac{\mathrm{d}{z}}{{z}} \\ $$

Question Number 221132 Answers: 2 Comments: 5

Question Number 221117 Answers: 3 Comments: 1

Question Number 221112 Answers: 0 Comments: 2

Question Number 221102 Answers: 3 Comments: 0

Question Number 221103 Answers: 0 Comments: 0

Prove : ∀x∈IR, ∀n∈IN^∗ ∫^( (π/2)) _( 0) ch(2xt)cos^(2n) (t) dt ≤ e^(x^2 /n) ∫^( (π/2)) _( 0) cos^(2n) (t) dt

$$\mathrm{Prove}\::\:\:\:\:\:\forall\mathrm{x}\in\mathrm{IR},\:\forall\mathrm{n}\in\mathrm{IN}^{\ast} \: \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ch}\left(\mathrm{2xt}\right)\mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt}\:\leqslant\:\mathrm{e}^{\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{n}}} \underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \mathrm{cos}^{\mathrm{2n}} \left(\mathrm{t}\right)\:\mathrm{dt} \\ $$

Question Number 221100 Answers: 0 Comments: 0

Prove; ∫_0 ^( +∞) ((4∙cos x ∙ ((sinh x ))^(1/(6 )) )/(sinh x + sinh 3x + 4 sinh^2 x − 2 sinh^2 2x + 4 sinh^4 x + 4 cosh^4 x)) dx = (𝛑/( (√6) + 2))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}; \\ $$$$\:\:\int_{\mathrm{0}} ^{\:+\infty} \:\frac{\mathrm{4}\centerdot\boldsymbol{\mathrm{cos}}\:\boldsymbol{{x}}\:\centerdot\:\sqrt[{\mathrm{6}\:\:}]{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:}}{\boldsymbol{\mathrm{sinh}}\:\boldsymbol{{x}}\:+\:\boldsymbol{\mathrm{sinh}}\:\mathrm{3}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\boldsymbol{{x}}\:−\:\mathrm{2}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{2}} \:\mathrm{2}\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{sinh}}^{\mathrm{4}} \:\boldsymbol{{x}}\:+\:\mathrm{4}\:\boldsymbol{\mathrm{cosh}}^{\mathrm{4}} \boldsymbol{{x}}}\:\boldsymbol{\mathrm{d}{x}}\:=\:\frac{\boldsymbol{\pi}}{\:\sqrt{\mathrm{6}}\:+\:\mathrm{2}}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 221099 Answers: 1 Comments: 0

Prove: ∫_( −π) ^( π) Σ_(n=0) ^∞ ((cos^(n + 1) x)/((n + 1)(1 + e^x^(2n +1) ))) dx = π ln2

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}: \\ $$$$\:\:\underset{\:−\pi} {\overset{\:\pi} {\int}}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{cos}^{{n}\:+\:\mathrm{1}} \:{x}}{\left({n}\:+\:\mathrm{1}\right)\left(\mathrm{1}\:+\:{e}^{{x}^{\mathrm{2}{n}\:+\mathrm{1}} } \right)}\:\:\mathrm{d}{x}\:=\:\pi\:\mathrm{ln2}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 221095 Answers: 3 Comments: 0

Complex integral 1. ∫_(−∞) ^( +∞) (dz/((z^2 +1)^ν ))=?? 2. ∫_(−∞) ^(+∞) (e^(iπt) /(t^2 +1)) dt=?? 3. ∮_( C) (1/z) dz=?? , C;x^2 +y^2 =1

$$\mathrm{Complex}\:\mathrm{integral} \\ $$$$\mathrm{1}.\:\int_{−\infty} ^{\:+\infty} \:\:\:\frac{\mathrm{d}{z}}{\left({z}^{\mathrm{2}} +\mathrm{1}\right)^{\nu} }=?? \\ $$$$\mathrm{2}.\:\int_{−\infty} ^{+\infty} \:\:\frac{{e}^{\boldsymbol{{i}}\pi{t}} }{{t}^{\mathrm{2}} +\mathrm{1}}\:\mathrm{d}{t}=?? \\ $$$$\mathrm{3}.\:\oint_{\:{C}} \:\frac{\mathrm{1}}{{z}}\:\mathrm{d}{z}=??\:,\:{C};{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{1} \\ $$

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