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Question Number 210308 Answers: 3 Comments: 1

Evaluate ∫((2y^4 )/(y^3 −y^2 +y−1))dy

$${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\int\frac{\mathrm{2}{y}^{\mathrm{4}} }{{y}^{\mathrm{3}} −{y}^{\mathrm{2}} +{y}−\mathrm{1}}{dy} \\ $$

Question Number 210307 Answers: 3 Comments: 0

∫((x^2 −1)/((x^2 +1)((√(1+x^4 )) )))

$$\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }\:\right)} \\ $$$$ \\ $$

Question Number 210297 Answers: 0 Comments: 0

Prove the theorem. A non empty subset W of a vector space V(F) is the subset of V if and only if αW_1 +βW_2 ∈W ∀α,β ∈ F and W_1 ,W_2 ∈W

$${Prove}\:{the}\:{theorem}. \\ $$$${A}\:{non}\:{empty}\:{subset}\:{W}\:\:{of}\:{a}\:{vector}\:{space}\:{V}\left({F}\right) \\ $$$${is}\:{the}\:{subset}\:{of}\:{V}\:\:{if}\:\:{and}\:{only}\:{if} \\ $$$$\alpha{W}_{\mathrm{1}} +\beta{W}_{\mathrm{2}} \:\in{W}\:\:\forall\alpha,\beta\:\in\:{F}\:\:{and}\:{W}_{\mathrm{1}} ,{W}_{\mathrm{2}} \:\in{W} \\ $$

Question Number 210298 Answers: 0 Comments: 0

For the given system of simultaneous linear equation 2x_1 −2x_2 +3x_3 +4x_4 −x_5 =0 −x_3 −2x_4 +3x_5 =0 −x_1 +x_2 +2x_3 +5x_4 +2x_5 =0 x_1 −x_2 +2x_3 +3x_4 =0 (a)Write the augmented matrix and convert it into echelon form (b)Hence find all the solution

$${For}\:{the}\:{given}\:{system}\:{of}\:{simultaneous}\: \\ $$$${linear}\:{equation} \\ $$$$\mathrm{2}{x}_{\mathrm{1}} −\mathrm{2}{x}_{\mathrm{2}} +\mathrm{3}{x}_{\mathrm{3}} +\mathrm{4}{x}_{\mathrm{4}} −{x}_{\mathrm{5}} =\mathrm{0} \\ $$$$−{x}_{\mathrm{3}} −\mathrm{2}{x}_{\mathrm{4}} +\mathrm{3}{x}_{\mathrm{5}} =\mathrm{0} \\ $$$$−{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +\mathrm{2}{x}_{\mathrm{3}} +\mathrm{5}{x}_{\mathrm{4}} +\mathrm{2}{x}_{\mathrm{5}} =\mathrm{0} \\ $$$${x}_{\mathrm{1}} −{x}_{\mathrm{2}} +\mathrm{2}{x}_{\mathrm{3}} +\mathrm{3}{x}_{\mathrm{4}} =\mathrm{0} \\ $$$$\left({a}\right){Write}\:{the}\:{augmented}\:\:{matrix}\:{and}\:{convert} \\ $$$${it}\:{into}\:{echelon}\:{form} \\ $$$$\left({b}\right){Hence}\:{find}\:{all}\:{the}\:{solution} \\ $$$$ \\ $$

Question Number 210295 Answers: 0 Comments: 1

Question Number 210289 Answers: 1 Comments: 0

Question Number 210263 Answers: 2 Comments: 0

Question Number 210261 Answers: 2 Comments: 0

show that ((sinAcosA−sinBcosB)/(cos^2 A−sin^2 B))=tan(A−B)

$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{\mathrm{sinAcosA}}−\boldsymbol{\mathrm{sinBcosB}}}{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{B}}}=\boldsymbol{\mathrm{tan}}\left(\boldsymbol{\mathrm{A}}−\boldsymbol{\mathrm{B}}\right) \\ $$

Question Number 210235 Answers: 2 Comments: 2

Question Number 210234 Answers: 2 Comments: 2

Question Number 210231 Answers: 0 Comments: 1

Resoudre dans R { ((acos x−bsin x=c (x≠0))),((sin ((1/(sin x))) =d (−1≤d≤+1))) :}

$$\mathrm{Resoudre}\:\boldsymbol{\mathrm{dans}}\:\mathbb{R} \\ $$$$\begin{cases}{\boldsymbol{\mathrm{a}}\mathrm{cos}\:\boldsymbol{\mathrm{x}}−\boldsymbol{\mathrm{b}}\mathrm{sin}\:\boldsymbol{\mathrm{x}}=\boldsymbol{\mathrm{c}}\:\:\:\:\:\left(\boldsymbol{\mathrm{x}}\neq\mathrm{0}\right)}\\{\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{sin}\:\boldsymbol{\mathrm{x}}}\right)\:\:\:\:\:\:\:\:\:=\boldsymbol{\mathrm{d}}\:\:\:\:\left(−\mathrm{1}\leqslant\boldsymbol{\mathrm{d}}\leqslant+\mathrm{1}\right)}\end{cases} \\ $$$$ \\ $$

Question Number 210208 Answers: 4 Comments: 0

∫_0 ^1 e^e^e^x e^e^x e^x dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{{e}^{{e}^{{x}} } } \:{e}^{{e}^{{x}} } \:{e}^{{x}} {dx} \\ $$$$ \\ $$

Question Number 210206 Answers: 0 Comments: 0

Ω=∫_(1/e) ^e (dx/((1+x^2 )(1+xlog^7 x)))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\frac{\mathrm{1}}{{e}}} ^{{e}} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\mathrm{log}\:^{\mathrm{7}} {x}\right)} \\ $$$$ \\ $$

Question Number 210229 Answers: 3 Comments: 0

Question Number 210369 Answers: 0 Comments: 0

Question Number 210171 Answers: 0 Comments: 0

Find: lim_(n→+∞) (n/((n!)^2 4^n )) Π_(k=1) ^n ((2k−1)^2 + 4) = ?

$$\mathrm{Find}: \\ $$$$\underset{\boldsymbol{\mathrm{n}}\rightarrow+\infty} {\mathrm{lim}}\:\:\frac{\mathrm{n}}{\left(\mathrm{n}!\right)^{\mathrm{2}} \:\mathrm{4}^{\boldsymbol{\mathrm{n}}} }\:\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\prod}}\:\left(\left(\mathrm{2k}−\mathrm{1}\right)^{\mathrm{2}} \:+\:\mathrm{4}\right)\:=\:? \\ $$

Question Number 210157 Answers: 3 Comments: 0

Question Number 210156 Answers: 1 Comments: 0

Question Number 210155 Answers: 1 Comments: 0

Question Number 210142 Answers: 0 Comments: 1

Question Number 210127 Answers: 1 Comments: 0

Question Number 210126 Answers: 1 Comments: 0

Question Number 210124 Answers: 0 Comments: 0

Question Number 210091 Answers: 1 Comments: 2

find Σ_(n=1) ^∞ tan^(−1) ((1/(2n^2 )))=?

$${find}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{2}{n}^{\mathrm{2}} }\right)=? \\ $$

Question Number 210081 Answers: 0 Comments: 0

Reduce [(3,(−2),4,7),(2,1,0,(−3)),(2,8,(−8),2) ] into echelon form

$${Reduce}\:\: \\ $$$$ \\ $$$$\:\:\:\begin{bmatrix}{\mathrm{3}}&{−\mathrm{2}}&{\mathrm{4}}&{\mathrm{7}}\\{\mathrm{2}}&{\mathrm{1}}&{\mathrm{0}}&{−\mathrm{3}}\\{\mathrm{2}}&{\mathrm{8}}&{−\mathrm{8}}&{\mathrm{2}}\end{bmatrix}\:\:\:\:{into}\:{echelon}\:{form} \\ $$$$ \\ $$

Question Number 210080 Answers: 1 Comments: 4

Given that det [(a,b,c),(d,e,f),(g,h,i) ]=n find det [((d+2a),(e+2b),(f+2c)),((2a),(2b),(2c)),((4g),(4h),(4i)) ]

$${Given}\:{that}\:\:{det}\:\begin{bmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{{g}}&{{h}}&{{i}}\end{bmatrix}={n} \\ $$$$ \\ $$$${find}\:{det}\begin{bmatrix}{{d}+\mathrm{2}{a}}&{{e}+\mathrm{2}{b}}&{{f}+\mathrm{2}{c}}\\{\mathrm{2}{a}}&{\mathrm{2}{b}}&{\mathrm{2}{c}}\\{\mathrm{4}{g}}&{\mathrm{4}{h}}&{\mathrm{4}{i}}\end{bmatrix} \\ $$$$ \\ $$

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