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Question Number 85858    Answers: 0   Comments: 0

Is a matrix A^T A always positive definite?

$$\mathrm{Is}\:\mathrm{a}\:\mathrm{matrix} \\ $$$$\mathrm{A}^{\mathrm{T}} \mathrm{A}\:\mathrm{always}\:\mathrm{positive}\:\mathrm{definite}? \\ $$

Question Number 85857    Answers: 1   Comments: 0

Question Number 85854    Answers: 0   Comments: 2

If x,y,z ∈ R satisfy the equation x^4 + y^4 + z^4 = 4xyz −1 find minimum value of x + y + z

$$\mathrm{If}\:\mathrm{x},\mathrm{y},\mathrm{z}\:\in\:\mathbb{R}\:\mathrm{satisfy}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:=\:\mathrm{4xyz}\:−\mathrm{1}\: \\ $$$$\mathrm{find}\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\: \\ $$

Question Number 85846    Answers: 0   Comments: 2

∫_( 0) ^(50π) ∣ cos x ∣dx =

$$\underset{\:\mathrm{0}} {\overset{\mathrm{50}\pi} {\int}}\:\mid\:\mathrm{cos}\:{x}\:\mid{dx}\:= \\ $$

Question Number 85845    Answers: 2   Comments: 0

solve tanh (x) = (1/(cosh (x)))

$$\mathrm{solve}\:\mathrm{tanh}\:\left(\mathrm{x}\right)\:=\:\frac{\mathrm{1}}{\mathrm{cosh}\:\left(\mathrm{x}\right)} \\ $$

Question Number 85839    Answers: 1   Comments: 1

∫x×(1/(√(x^2 −1)))dx

$$\int\mathrm{x}×\frac{\mathrm{1}}{\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\mathrm{dx} \\ $$

Question Number 85835    Answers: 1   Comments: 0

xydy=(y^2 +x)dx

$$\mathrm{xydy}=\left(\mathrm{y}^{\mathrm{2}} +\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 85834    Answers: 1   Comments: 0

2y^′ −(x/y)=((xy)/(x^2 −1))

$$\mathrm{2y}^{'} −\frac{\mathrm{x}}{\mathrm{y}}=\frac{\mathrm{xy}}{\mathrm{x}^{\mathrm{2}} −\mathrm{1}} \\ $$

Question Number 85832    Answers: 1   Comments: 1

(x+x^(−1) )^2 +(x^2 +x^(−2) )^2 +(x^3 +x^(−3) )^2 + ... + (x^(10) +x^(−10) )^2 =

$$\left(\mathrm{x}+\mathrm{x}^{−\mathrm{1}} \right)^{\mathrm{2}} +\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}^{−\mathrm{2}} \right)^{\mathrm{2}} +\left(\mathrm{x}^{\mathrm{3}} +\mathrm{x}^{−\mathrm{3}} \right)^{\mathrm{2}} \\ $$$$+\:...\:+\:\left(\mathrm{x}^{\mathrm{10}} +\mathrm{x}^{−\mathrm{10}} \right)^{\mathrm{2}} \:=\: \\ $$

Question Number 85828    Answers: 1   Comments: 0

∫(1/(x+cot(x))) dx

$$\int\frac{\mathrm{1}}{{x}+{cot}\left({x}\right)}\:{dx} \\ $$

Question Number 85826    Answers: 1   Comments: 2

^x log (xy).^y log (xy) +^x log (x−y).^y log (x−y)=0 find x+y

$$\:^{\mathrm{x}} \mathrm{log}\:\left(\mathrm{xy}\right).\:^{\mathrm{y}} \mathrm{log}\:\left(\mathrm{xy}\right)\:+\:^{\mathrm{x}} \mathrm{log}\:\left(\mathrm{x}−\mathrm{y}\right).^{\mathrm{y}} \mathrm{log}\:\left(\mathrm{x}−\mathrm{y}\right)=\mathrm{0} \\ $$$$\mathrm{find}\:\mathrm{x}+\mathrm{y}\: \\ $$

Question Number 85822    Answers: 2   Comments: 1

how to solve ((x−1))^(1/(3 )) + ((x−3))^(1/(3 )) + ((x−5))^(1/(3 )) = 0

$$\mathrm{how}\:\mathrm{to}\:\mathrm{solve}\: \\ $$$$\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{1}}\:+\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{3}}\:+\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}−\mathrm{5}}\:=\:\mathrm{0}\: \\ $$

Question Number 85817    Answers: 2   Comments: 0

what is coefficient of x^2 in the expansion [ (1−x)(1+2x)]^6

$$\mathrm{what}\:\mathrm{is}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{2}} \:\mathrm{in}\: \\ $$$$\mathrm{the}\:\mathrm{expansion}\:\left[\:\left(\mathrm{1}−\mathrm{x}\right)\left(\mathrm{1}+\mathrm{2x}\right)\right]^{\mathrm{6}} \\ $$

Question Number 85807    Answers: 1   Comments: 0

∫_0 ^1 ((x^2 dx)/(√(1−x^4 )))

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{2}} \:\mathrm{dx}}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }} \\ $$

Question Number 85801    Answers: 0   Comments: 3

calculate ∫_0 ^π (dx/((cosx +2sinx)^2 ))

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{dx}}{\left({cosx}\:+\mathrm{2}{sinx}\right)^{\mathrm{2}} } \\ $$

Question Number 85793    Answers: 0   Comments: 0

∫_1 ^2 ((tan^(−1) (x−1) ln(x−1))/x) dx

$$\int_{\mathrm{1}} ^{\mathrm{2}} \frac{{tan}^{−\mathrm{1}} \left({x}−\mathrm{1}\right)\:{ln}\left({x}−\mathrm{1}\right)}{{x}}\:{dx} \\ $$

Question Number 85789    Answers: 1   Comments: 0

∫(ln x)^2 dx =

$$\int\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} \:{dx}\:= \\ $$

Question Number 85786    Answers: 0   Comments: 0

posons (1+2(√3))^n =a_n +b_n (√3) montre que pgcd(a_n ;b_n )=1

$${posons}\: \\ $$$$\left(\mathrm{1}+\mathrm{2}\sqrt{\mathrm{3}}\right)^{\boldsymbol{{n}}} =\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} +\boldsymbol{\mathrm{b}}_{\boldsymbol{\mathrm{n}}} \sqrt{\mathrm{3}} \\ $$$$\boldsymbol{\mathrm{montre}}\:\boldsymbol{\mathrm{que}}\:\boldsymbol{\mathrm{pgcd}}\left(\boldsymbol{\mathrm{a}}_{\boldsymbol{\mathrm{n}}} ;\boldsymbol{{b}}_{\boldsymbol{{n}}} \right)=\mathrm{1} \\ $$

Question Number 85781    Answers: 1   Comments: 2

∫_0 ^1 (ln (1/x))^(−3/2) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(\mathrm{ln}\:\frac{\mathrm{1}}{{x}}\right)^{−\mathrm{3}/\mathrm{2}} \:{dx} \\ $$

Question Number 85779    Answers: 0   Comments: 0

Question Number 85776    Answers: 1   Comments: 0

Question Number 85775    Answers: 0   Comments: 0

Question Number 85774    Answers: 0   Comments: 0

(x_(2n) )=2^(2n) ((x/2))_n (((x+1)/2))_n (x)_(m n) =m^(m n) Π_(k=0) ^(m=1) (((x+k)/m))_n , m∈z now if m is relative number such as(3/2) , m∈Q (x)_((3/2)n) =?? help me

$$\left({x}_{\mathrm{2}{n}} \right)=\mathrm{2}^{\mathrm{2}{n}} \left(\frac{{x}}{\mathrm{2}}\right)_{{n}} \left(\frac{{x}+\mathrm{1}}{\mathrm{2}}\right)_{{n}} \\ $$$$\left({x}\right)_{{m}\:{n}} ={m}^{{m}\:{n}} \underset{{k}=\mathrm{0}} {\overset{{m}=\mathrm{1}} {\prod}}\left(\frac{{x}+{k}}{{m}}\right)_{{n}} \:\:\:,\:{m}\in{z} \\ $$$${now}\:{if}\:{m}\:{is}\:{relative}\:{number}\:{such}\:{as}\frac{\mathrm{3}}{\mathrm{2}}\:,\:{m}\in{Q} \\ $$$$\left({x}\right)_{\frac{\mathrm{3}}{\mathrm{2}}{n}} =?? \\ $$$$ \\ $$$${help}\:{me}\: \\ $$

Question Number 85766    Answers: 0   Comments: 2

(dy/dx) = 1−sin (x+2y)

$$\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{1}−\mathrm{sin}\:\left(\mathrm{x}+\mathrm{2y}\right) \\ $$

Question Number 85762    Answers: 0   Comments: 12

Question Number 85760    Answers: 0   Comments: 0

∫(([cos^(−1) (x){(√(1−x^2 ))}]^(−1) )/(log{((sin(2x(√(1−x^2 ))))/π)})) dx

$$\int\frac{\left[{cos}^{−\mathrm{1}} \left({x}\right)\left\{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right\}\right]^{−\mathrm{1}} }{{log}\left\{\frac{{sin}\left(\mathrm{2}{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\right)}{\pi}\right\}}\:{dx} \\ $$

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