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Question Number 189394    Answers: 0   Comments: 0

Prove that: ln(n+1)<(1/( (√(1^2 +1))))+(1/( (√(2^2 +2))))+...+(1/( (√(n^2 +n))))(∀n∈N^∗ )

$${Prove}\:{that}: \\ $$$${ln}\left({n}+\mathrm{1}\right)<\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}^{\mathrm{2}} +\mathrm{1}}}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}^{\mathrm{2}} +\mathrm{2}}}+...+\frac{\mathrm{1}}{\:\sqrt{{n}^{\mathrm{2}} +{n}}}\left(\forall{n}\in{N}^{\ast} \right) \\ $$

Question Number 189390    Answers: 0   Comments: 0

Question Number 189384    Answers: 0   Comments: 0

Question Number 189382    Answers: 1   Comments: 0

Question Number 189383    Answers: 0   Comments: 0

Question Number 189375    Answers: 3   Comments: 0

show that : (1/(cscx + cot x)) = cscx − cot x

$$ \\ $$$$\:\:\:\:{show}\:{that}\:: \\ $$$$\:\:\:\:\frac{\mathrm{1}}{{cscx}\:+\:{cot}\:{x}}\:=\:{cscx}\:−\:{cot}\:{x} \\ $$$$ \\ $$$$ \\ $$

Question Number 189374    Answers: 1   Comments: 0

Δ={(x,y,z) ∈ R^3 :x^2 +y^2 ≤1 , x≥0, 0≤z≤1+y}. calculate: ∫∫∫_Δ dxdydz.

$$\Delta=\left\{\left({x},{y},{z}\right)\:\in\:\mathbb{R}^{\mathrm{3}} :{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \leqslant\mathrm{1}\:,\:\:{x}\geqslant\mathrm{0},\:\mathrm{0}\leqslant{z}\leqslant\mathrm{1}+{y}\right\}. \\ $$$${calculate}: \\ $$$$\int\int\int_{\Delta} {dxdydz}. \\ $$

Question Number 189436    Answers: 1   Comments: 1

Question Number 189438    Answers: 1   Comments: 0

Question Number 189367    Answers: 1   Comments: 0

Question Number 189357    Answers: 2   Comments: 5

How many pairs of positive integers x, y exist such that HCF (x, y) + LCM(x, y) = 91?

$$ \\ $$How many pairs of positive integers x, y exist such that HCF (x, y) + LCM(x, y) = 91?

Question Number 189350    Answers: 1   Comments: 0

Question Number 189345    Answers: 1   Comments: 0

solve ∫t^(−6) (t^2 +3)^2 dt

$${solve} \\ $$$$\int{t}^{−\mathrm{6}} \left({t}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{2}} {dt} \\ $$

Question Number 189339    Answers: 1   Comments: 0

Question Number 189363    Answers: 2   Comments: 0

Question Number 189335    Answers: 0   Comments: 0

determine the volume of the region that is between the xy plane and f(x,y)=2+cos(x^2 ) and is above the triangle with vertices (0,0),(6,0) and (6,2) using double integral

$$\boldsymbol{\mathrm{determine}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{volume}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{region}}\: \\ $$$$\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{between}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{xy}}\:\boldsymbol{\mathrm{plane}} \\ $$$$\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\right)=\mathrm{2}+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{above}} \\ $$$$\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{triangle}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{vertices}}\:\left(\mathrm{0},\mathrm{0}\right),\left(\mathrm{6},\mathrm{0}\right)\: \\ $$$${a}\boldsymbol{{nd}}\:\left(\mathrm{6},\mathrm{2}\right)\:\boldsymbol{{using}}\:\boldsymbol{{double}}\:\boldsymbol{{integral}} \\ $$

Question Number 189325    Answers: 2   Comments: 0

prove Ω= ∫_0 ^(π/2) (( cos(x)+cos(5x))/(1+ 2sin(x))) =^( ?) (3/2)

$$ \\ $$$$\:\:\:\:\:{prove} \\ $$$$ \\ $$$$\:\:\Omega=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\:{cos}\left({x}\right)+{cos}\left(\mathrm{5}{x}\right)}{\mathrm{1}+\:\mathrm{2}{sin}\left({x}\right)}\:\overset{\:?} {=}\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$ \\ $$

Question Number 189323    Answers: 1   Comments: 0

Question Number 189319    Answers: 0   Comments: 4

Question Number 189309    Answers: 1   Comments: 0

Question Number 189304    Answers: 2   Comments: 1

Question Number 189302    Answers: 1   Comments: 2

Q: find the number of the solutions for : ( x_( 1) + x_( 2) )^( 3) + x_( 3) + x_( 4) + x_( 5) =11 Hint: ( x_( i) ∈ Z^( +) ∪ { 0 } )

$$ \\ $$$$\:\:\:\:{Q}:\:\:\:\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{the}\:\:\mathrm{solutions}\:\:\mathrm{for}\:: \\ $$$$ \\ $$$$\:\:\left(\:{x}_{\:\mathrm{1}} \:+\:{x}_{\:\mathrm{2}} \:\right)^{\:\mathrm{3}} \:+\:{x}_{\:\mathrm{3}} \:+\:{x}_{\:\mathrm{4}} \:+\:{x}_{\:\mathrm{5}} \:=\mathrm{11} \\ $$$$\:\:\: \\ $$$$\:\:\:\:{Hint}:\:\:\:\left(\:{x}_{\:{i}} \:\:\in\:\:\mathbb{Z}^{\:\:+} \:\:\cup\:\left\{\:\mathrm{0}\:\right\}\:\:\right) \\ $$$$\: \\ $$

Question Number 189293    Answers: 1   Comments: 0

Question Number 189292    Answers: 0   Comments: 2

Question Number 189291    Answers: 0   Comments: 0

Question Number 189290    Answers: 1   Comments: 0

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