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Question Number 88811 Answers: 1 Comments: 0

{ (((x+1)^2 (y+1)^2 =27xy)),(((x^2 +1)(y^2 +1) =10xy)) :}

$$\begin{cases}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({y}+\mathrm{1}\right)^{\mathrm{2}} =\mathrm{27}{xy}}\\{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({y}^{\mathrm{2}} +\mathrm{1}\right)\:=\mathrm{10}{xy}}\end{cases} \\ $$

Question Number 88789 Answers: 0 Comments: 7

∫∫ln(x+1) dx dy

$$\int\int{ln}\left({x}+\mathrm{1}\right)\:{dx}\:{dy} \\ $$

Question Number 88778 Answers: 0 Comments: 3

Question Number 88794 Answers: 1 Comments: 0

Question Number 88771 Answers: 0 Comments: 4

solve x^x^4 =64

$${solve} \\ $$$${x}^{{x}^{\mathrm{4}} } =\mathrm{64} \\ $$

Question Number 88761 Answers: 0 Comments: 4

if a_n =((n!)/(n^n e^(−n) (√(2πn)))) and b_n =(((2n)!(√n))/(4^n (n!)^2 )) lim_(n→∞) a_n =1 find lim_(n→∞) b_n =?

$${if}\:{a}_{{n}} =\frac{{n}!}{{n}^{{n}} \:{e}^{−{n}} \sqrt{\mathrm{2}\pi{n}}} \\ $$$${and}\:{b}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!\sqrt{{n}}}{\mathrm{4}^{{n}} \:\left({n}!\right)^{\mathrm{2}} } \\ $$$$\underset{{n}\rightarrow\infty} {{lim}a}_{{n}} =\mathrm{1} \\ $$$${find}\:\underset{{n}\rightarrow\infty} {{lim}b}_{{n}} =? \\ $$

Question Number 88758 Answers: 2 Comments: 22

Some people may have noticed that i usually calculate areas concerning parabola directly, without applying complicated integral calculus. Here i am giving you the backgroud. Actually you know all these things and you are able to prove them. Maybe you just forget to apply them.

$${Some}\:{people}\:{may}\:{have}\:{noticed}\:{that}\:{i} \\ $$$${usually}\:{calculate}\:{areas}\:{concerning} \\ $$$${parabola}\:{directly},\:{without}\:{applying} \\ $$$${complicated}\:{integral}\:{calculus}. \\ $$$${Here}\:{i}\:{am}\:{giving}\:{you}\:{the}\:{backgroud}.\: \\ $$$${Actually}\:{you}\:{know}\:{all}\:{these}\:{things}\:{and} \\ $$$${you}\:{are}\:{able}\:{to}\:{prove}\:{them}.\:{Maybe} \\ $$$${you}\:{just}\:{forget}\:{to}\:{apply}\:{them}. \\ $$

Question Number 88756 Answers: 0 Comments: 0

E is reported in (i^→ ;j^→ ) base. e_1 ^→ =2i^→ +3j^→ ; e_2 ^→ =i^→ −2j^→ and e_3 ^→ =4i^→ −5j^→ belong to E. 1)Determinate the cordonnates of e_3 ^→ in the base B(e_1 ^→ ;e_2 ^→ ).

$$\mathrm{E}\:\mathrm{is}\:\mathrm{reported}\:\mathrm{in}\:\left(\overset{\rightarrow} {\mathrm{i}};\overset{\rightarrow} {\mathrm{j}}\right)\:\mathrm{base}. \\ $$$$\overset{\rightarrow} {\mathrm{e}}_{\mathrm{1}} =\mathrm{2}\overset{\rightarrow} {\mathrm{i}}+\mathrm{3}\overset{\rightarrow} {\mathrm{j}}\:;\:\:\:\overset{\rightarrow} {\mathrm{e}}_{\mathrm{2}} =\overset{\rightarrow} {\mathrm{i}}−\mathrm{2}\overset{\rightarrow} {\mathrm{j}}\:\mathrm{and}\:\:\:\overset{\rightarrow} {\mathrm{e}}_{\mathrm{3}} =\mathrm{4}\overset{\rightarrow} {\mathrm{i}}−\mathrm{5}\overset{\rightarrow} {\mathrm{j}}\: \\ $$$$\mathrm{belong}\:\mathrm{to}\:\mathrm{E}. \\ $$$$\left.\mathrm{1}\right)\mathrm{Determinate}\:\mathrm{the}\:\mathrm{cordonnates}\:\mathrm{of}\:\overset{\rightarrow} {\mathrm{e}}_{\mathrm{3}} \:\mathrm{in}\:\mathrm{the}\: \\ $$$$\mathrm{base}\:\mathrm{B}\left(\overset{\rightarrow} {\mathrm{e}}_{\mathrm{1}} ;\overset{\rightarrow} {\mathrm{e}}_{\mathrm{2}} \right). \\ $$

Question Number 88754 Answers: 1 Comments: 3

Question Number 88752 Answers: 1 Comments: 2

cos(𝛂)+cos(𝛃)+cos(𝛄)≤(3/2) prove the inequality

$$\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\alpha}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\beta}\right)+\boldsymbol{\mathrm{cos}}\left(\boldsymbol{\gamma}\right)\leqslant\frac{\mathrm{3}}{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{prove}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{inequality}} \\ $$

Question Number 88731 Answers: 0 Comments: 3

Question Number 88723 Answers: 1 Comments: 0

prove that Σ_(k=0) ^∞ (((k+2)^2 x^k )/((k+3)!))=(e^x /x^3 )(x^2 −x+1)−((x^2 +2)/(2x^3 ))

$${prove}\:{that} \\ $$$$ \\ $$$$\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left({k}+\mathrm{2}\right)^{\mathrm{2}} {x}^{{k}} }{\left({k}+\mathrm{3}\right)!}=\frac{{e}^{{x}} }{{x}^{\mathrm{3}} }\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)−\frac{{x}^{\mathrm{2}} +\mathrm{2}}{\mathrm{2}{x}^{\mathrm{3}} } \\ $$

Question Number 88710 Answers: 2 Comments: 1

∫_(−(√3) ) ^(√3) ∫_1 ^(√(4−x^2 )) (x^2 +y^2 )^(3/2) dydx

$$\underset{−\sqrt{\mathrm{3}}\:} {\overset{\sqrt{\mathrm{3}}} {\int}}\underset{\mathrm{1}} {\overset{\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }} {\int}}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)^{\mathrm{3}/\mathrm{2}} \:{dydx} \\ $$

Question Number 88708 Answers: 2 Comments: 5

Question Number 88690 Answers: 1 Comments: 0

Question Number 88683 Answers: 2 Comments: 0

Question Number 88681 Answers: 0 Comments: 2

If f(x) is a periodic function, with period T, then

$$\mathrm{If}\:\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{periodic}\:\mathrm{function},\:\mathrm{with} \\ $$$$\mathrm{period}\:{T},\:\mathrm{then} \\ $$

Question Number 88678 Answers: 0 Comments: 8

Find (√i)+(√(−i))

$$\boldsymbol{\mathrm{F}}{ind}\:\:\:\sqrt{\boldsymbol{{i}}}+\sqrt{−\boldsymbol{\mathrm{i}}} \\ $$

Question Number 88673 Answers: 2 Comments: 2

Question Number 88663 Answers: 1 Comments: 0

{ ((log_2 x+log_4 y=4)),((x.y=8)) :}

$$\begin{cases}{{log}_{\mathrm{2}} {x}+{log}_{\mathrm{4}} {y}=\mathrm{4}}\\{{x}.{y}=\mathrm{8}}\end{cases} \\ $$

Question Number 88656 Answers: 1 Comments: 1

Question Number 88642 Answers: 1 Comments: 2

Question Number 88623 Answers: 2 Comments: 4

Question Number 88616 Answers: 1 Comments: 9

Question Number 88613 Answers: 1 Comments: 3

Question Number 88611 Answers: 0 Comments: 0

prove that ((1+p^2 +p^4 +......+p^(2n) )/(p+p^3 +p^5 +.....p^(2n−1) ))>((n+1)/(np))

$${prove}\:{that} \\ $$$$\frac{\mathrm{1}+{p}^{\mathrm{2}} +{p}^{\mathrm{4}} +......+{p}^{\mathrm{2}{n}} }{{p}+{p}^{\mathrm{3}} +{p}^{\mathrm{5}} +.....{p}^{\mathrm{2}{n}−\mathrm{1}} }>\frac{{n}+\mathrm{1}}{{np}} \\ $$

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