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Question Number 220958 Answers: 1 Comments: 1

for x, y, z >0 find the maximum of x^m y^n z^k subject to ax+by+cz=d.

$${for}\:{x},\:{y},\:{z}\:>\mathrm{0}\:{find}\:{the}\:{maximum}\:{of} \\ $$$${x}^{{m}} {y}^{{n}} {z}^{{k}} \:{subject}\:{to}\:{ax}+{by}+{cz}={d}. \\ $$

Question Number 220950 Answers: 1 Comments: 0

∫_( 0) ^( π) ∫_( 0) ^( 1) ∫_( 0) ^( π) sin^( 2) x + y sin z dxdydz = (1/2) π (2 + π)

$$ \\ $$$$\:\:\:\:\int_{\:\mathrm{0}} ^{\:\pi} \int_{\:\mathrm{0}} ^{\:\mathrm{1}} \int_{\:\mathrm{0}} ^{\:\:\pi} \:\mathrm{sin}^{\:\mathrm{2}} \:{x}\:+\:{y}\:\mathrm{sin}\:{z}\:{dxdydz}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\pi\:\left(\mathrm{2}\:+\:\pi\right)\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220948 Answers: 1 Comments: 0

∫ x^2 (√(5−x^6 ))dx

$$\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{5}−{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 220947 Answers: 1 Comments: 0

Σ_(k=1) ^(13) (1/(sin ((π/4)+(((k−1)π)/6))sin ((π/4)+((kπ)/6))))

$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{13}} {\sum}}\:\:\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{\left({k}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{{k}\pi}{\mathrm{6}}\right)} \\ $$

Question Number 220913 Answers: 0 Comments: 4

Is there an Manager??? pls ban Question Spamming and... pls fix invisible line matrix bug

$$\mathrm{Is}\:\mathrm{there}\:\mathrm{an}\:\mathrm{Manager}??? \\ $$$$\mathrm{pls}\:\mathrm{ban}\:\mathrm{Question}\:\mathrm{Spamming}\:\mathrm{and}... \\ $$$$\mathrm{pls}\:\mathrm{fix}\:\mathrm{invisible}\:\mathrm{line}\:\mathrm{matrix}\:\mathrm{bug} \\ $$

Question Number 220904 Answers: 0 Comments: 0

∫∫∫_([0,1]^3 ) ((x^4 y^3 z^2 )/((x+y+z)(x^2 +y^2 +z^2 )−(x^3 +y^3 +z^3 ))) dxdydz

$$ \\ $$$$\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{x}^{\mathrm{4}} {y}^{\mathrm{3}} {z}^{\mathrm{2}} }{\left({x}+{y}+{z}\right)\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} \right)−\left({x}^{\mathrm{3}} +{y}^{\mathrm{3}} +{z}^{\mathrm{3}} \right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$\: \\ $$

Question Number 220899 Answers: 0 Comments: 0

∫∫∫_([0,1]^( 3) ) (1/((1+x^2 )(1+y^2 )(1+z^2 )(1+xyz))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{y}^{\mathrm{2}} \right)\left(\mathrm{1}+{z}^{\mathrm{2}} \right)\left(\mathrm{1}+{xyz}\right)}\:{dxdydz}\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220898 Answers: 0 Comments: 0

∫∫∫_([0,1]^( 3) ) (1/( (√((1 −x)(1 − y)(1 −z)(1 − xyz))))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} \:\:} \frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}\:−{x}\right)\left(\mathrm{1}\:−\:{y}\right)\left(\mathrm{1}\:−{z}\right)\left(\mathrm{1}\:−\:{xyz}\right)}}\:{dxdydz}\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220897 Answers: 0 Comments: 0

∫∫∫_( [0,1]^( 3) ) (1/(1 + x^2 y^2 + y^2 z^2 + z^2 x^2 )) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\right]^{\:\mathrm{3}} } \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} {z}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} {x}^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220896 Answers: 1 Comments: 0

∫∫∫_( [0,∞]^( 3) ) ((x^2 y^2 z^2 )/((1 + x^2 + y^2 + z^2 )^5 )) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\:\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \frac{{x}^{\mathrm{2}} {y}^{\mathrm{2}} {z}^{\mathrm{2}} }{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{5}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220895 Answers: 0 Comments: 0

∫∫∫_([0,1]^3 ) ((ln (1 + xyz))/((1 + x)(1 + y)(1 + z))) dxdydz

$$ \\ $$$$\:\:\:\int\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{3}} } \frac{{ln}\:\left(\mathrm{1}\:+\:{xyz}\right)}{\left(\mathrm{1}\:+\:{x}\right)\left(\mathrm{1}\:+\:{y}\right)\left(\mathrm{1}\:+\:{z}\right)}\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220892 Answers: 2 Comments: 0

∫∫∫_(x^2 + y^2 + z^2 ≤ 1) (1/((1 + x^2 +y^2 + z^2 )^2 )) dxdydz

$$ \\ $$$$\:\:\:\:\int\int\int_{\boldsymbol{{x}}^{\mathrm{2}} \:+\:\boldsymbol{{y}}^{\mathrm{2}} \:+\:\boldsymbol{{z}}^{\mathrm{2}} \:\:\leqslant\:\mathrm{1}} \:\frac{\mathrm{1}}{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dxdydz}\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Question Number 220891 Answers: 1 Comments: 0

∫∫∫_([0,∞]^( 3) ) (e^(−(x + y + z )) /(1 + xyz)) dxdydz

$$ \\ $$$$\:\:\:\:\:\int\int\int_{\left[\mathrm{0},\infty\right]^{\:\mathrm{3}} } \:\frac{{e}^{−\left({x}\:+\:{y}\:+\:{z}\:\right)} }{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$

Question Number 220889 Answers: 2 Comments: 0

∫∫∫_( [0,1[^( 3) ) (1/(1 + xyz)) dxdydz

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\int\int\int_{\:\left[\mathrm{0},\mathrm{1}\left[^{\:\mathrm{3}} \right.\right.} \:\frac{\mathrm{1}}{\mathrm{1}\:+\:{xyz}}\:{dxdydz} \\ $$$$ \\ $$

Question Number 220878 Answers: 1 Comments: 0

Question Number 220877 Answers: 4 Comments: 0

Question Number 220876 Answers: 3 Comments: 0

Question Number 220874 Answers: 1 Comments: 2

Question Number 220873 Answers: 1 Comments: 0

Question Number 220872 Answers: 1 Comments: 0

Question Number 220869 Answers: 1 Comments: 1

Find the maximum value of x^2 y^3 z^4 subject to the condition x+y+z=18

$${Find}\:{the}\:{maximum}\:{value}\:{of}\:{x}^{\mathrm{2}} {y}^{\mathrm{3}} {z}^{\mathrm{4}} \:{subject}\:{to}\:{the}\:{condition}\:{x}+{y}+{z}=\mathrm{18} \\ $$

Question Number 220863 Answers: 1 Comments: 0

(211) Find the derivative of Δx, where Δx= determinant (((f_1 (x)),(φ_1 (x)),(Ψ_1 (x))),((f_2 (x)),(φ_2 (x)),(Ψ_2 (x))),((f_3 (x)),(φ_3 (x)),(Ψ_3 (x)))) and f_1 (x) ,f_2 (x), f_3 (x),φ_1 (x), etc. are different functions of x.

$$\left(\mathrm{211}\right) \\ $$$$\:\: \\ $$$${Find}\:{the}\:{derivative}\:{of}\:\Delta{x},\:{where} \\ $$$$\Delta{x}=\begin{vmatrix}{{f}_{\mathrm{1}} \left({x}\right)}&{\phi_{\mathrm{1}} \left({x}\right)}&{\Psi_{\mathrm{1}} \left({x}\right)}\\{{f}_{\mathrm{2}} \left({x}\right)}&{\phi_{\mathrm{2}} \left({x}\right)}&{\Psi_{\mathrm{2}} \left({x}\right)}\\{{f}_{\mathrm{3}} \left({x}\right)}&{\phi_{\mathrm{3}} \left({x}\right)}&{\Psi_{\mathrm{3}} \left({x}\right)}\end{vmatrix} \\ $$$${and}\:{f}_{\mathrm{1}} \left({x}\right)\:,{f}_{\mathrm{2}} \left({x}\right),\:{f}_{\mathrm{3}} \left({x}\right),\phi_{\mathrm{1}} \left({x}\right),\:{etc}.\:{are}\:{different}\:{functions}\:{of}\:{x}. \\ $$

Question Number 220858 Answers: 1 Comments: 2

Question Number 220857 Answers: 1 Comments: 0

Prove that tan 20^0 tan40^0 tan 80^0 =tan 60^0

$${Prove}\:{that}\:\mathrm{tan}\:\mathrm{20}^{\mathrm{0}} \mathrm{tan40}^{\mathrm{0}} \:\mathrm{tan}\:\mathrm{80}^{\mathrm{0}} =\mathrm{tan}\:\mathrm{60}^{\mathrm{0}} \\ $$

Question Number 220855 Answers: 1 Comments: 0

If b cos(θ+120^0 )=c cos (θ+240^0 ) then prove that b−c=−(b+c)(√3) tan θ

$${If}\:\:{b}\:\mathrm{cos}\left(\theta+\mathrm{120}^{\mathrm{0}} \right)={c}\:\mathrm{cos}\:\left(\theta+\mathrm{240}^{\mathrm{0}} \right)\:{then}\:{prove}\:{that} \\ $$$${b}−{c}=−\left({b}+{c}\right)\sqrt{\mathrm{3}}\:\mathrm{tan}\:\theta \\ $$

Question Number 220854 Answers: 2 Comments: 0

Solve for x and y 3^x +3^y =4, 3^(−x) +3^(−y ) =(4/3)

$${Solve}\:{for}\:{x}\:\:\:{and}\:\:\:\:{y} \\ $$$$\mathrm{3}^{{x}} +\mathrm{3}^{{y}} =\mathrm{4},\:\:\mathrm{3}^{−{x}} +\mathrm{3}^{−{y}\:} =\frac{\mathrm{4}}{\mathrm{3}} \\ $$

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