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Question Number 152947    Answers: 1   Comments: 0

prove :: š›—=āˆ«_(āˆ’āˆž) ^( āˆž) (( e^( āˆ’(1/x^( 2) )) )/x^( 4) ) dx =^? (1/2) Ī“ ((1/2) )

$$ \\ $$$$\:\:\:{prove}\::: \\ $$$$\:\:\:\boldsymbol{\phi}=\int_{āˆ’\infty} ^{\:\infty} \:\frac{\:{e}^{\:āˆ’\frac{\mathrm{1}}{{x}^{\:\mathrm{2}} }} }{{x}^{\:\mathrm{4}} }\:{dx}\:\overset{?} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\Gamma\:\left(\frac{\mathrm{1}}{\mathrm{2}}\:\right) \\ $$$$ \\ $$

Question Number 152946    Answers: 0   Comments: 1

((4038)/(1+(1/3)+(1/6)+(1/(10))+(1/(15))+...+(1/(1+2+3+4+...+2019)))) =?

$$\:\:\:\frac{\mathrm{4038}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{6}}+\frac{\mathrm{1}}{\mathrm{10}}+\frac{\mathrm{1}}{\mathrm{15}}+...+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+...+\mathrm{2019}}}\:=? \\ $$

Question Number 152942    Answers: 2   Comments: 1

Question Number 152940    Answers: 1   Comments: 0

In bottle manufacturing company, it was observed that 5% of the bottles manufactured were defective. In a random sample of 150 bottles, find probability that (a) exactly 3, (b) between 3 and 6, (c) at most 4, manufactured bottles are defective. [Take e = 2.718]

$$\:\mathrm{In}\:\mathrm{bottle}\:\mathrm{manufacturing}\:\mathrm{company},\:\mathrm{it} \\ $$$$\mathrm{was}\:\mathrm{observed}\:\mathrm{that}\:\mathrm{5\%}\:\mathrm{of}\:\mathrm{the}\:\mathrm{bottles} \\ $$$$\mathrm{manufactured}\:\mathrm{were}\:\mathrm{defective}.\:\mathrm{In}\:\mathrm{a}\: \\ $$$$\mathrm{random}\:\mathrm{sample}\:\mathrm{of}\:\mathrm{150}\:\mathrm{bottles},\:\mathrm{find}\: \\ $$$$\mathrm{probability}\:\mathrm{that}\: \\ $$$$\:\left({a}\right)\:\mathrm{exactly}\:\mathrm{3}, \\ $$$$\:\left({b}\right)\:\mathrm{between}\:\mathrm{3}\:\mathrm{and}\:\mathrm{6}, \\ $$$$\:\left({c}\right)\:\mathrm{at}\:\mathrm{most}\:\mathrm{4}, \\ $$$$\:\mathrm{manufactured}\:\mathrm{bottles}\:\mathrm{are}\:\mathrm{defective}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left[\mathrm{Take}\:\:{e}\:=\:\mathrm{2}.\mathrm{718}\right] \\ $$

Question Number 152939    Answers: 1   Comments: 0

Q : If a , b are positive numbers and { (( a = 1 + (( 6a āˆ’2))^(1/3) )),(( b = 1 + (( 6b āˆ’2))^(1/3) )) :} then find the value of , a.b =? ... Compiled by m.n : (E lementary olympiad ). ā– 

$$ \\ $$$$\:\:\:\:\:\mathrm{Q}\::\:\:\mathrm{If}\:\:\:\:{a}\:\:,\:\:{b}\:\:\:\:\mathrm{are}\:\mathrm{positive}\:\mathrm{numbers}\:\:\mathrm{and} \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\begin{cases}{\:\:{a}\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\mathrm{6}{a}\:āˆ’\mathrm{2}}\:\:}\\{\:\:\:{b}\:=\:\mathrm{1}\:+\:\sqrt[{\mathrm{3}}]{\:\mathrm{6}{b}\:āˆ’\mathrm{2}}}\end{cases} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:,\:\:\:\:{a}.{b}\:=? \\ $$$$\:\:\:\:...\:\mathrm{Compiled}\:\mathrm{by}\:\mathrm{m}.\mathrm{n}\::\:\left(\mathscr{E}\:{lementary}\:{olympiad}\:\right).\:\:\:\:\:\:\blacksquare \\ $$$$ \\ $$

Question Number 152937    Answers: 1   Comments: 0

Question Number 152935    Answers: 1   Comments: 0

Question Number 152918    Answers: 1   Comments: 0

Find the first derivative of y=x(āˆš(16āˆ’x^2 ))+16sin^(āˆ’1) (x/4)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{first}\:\mathrm{derivative}\:\mathrm{of}\: \\ $$$${y}={x}\sqrt{\mathrm{16}āˆ’{x}^{\mathrm{2}} }+\mathrm{16sin}^{āˆ’\mathrm{1}} \frac{{x}}{\mathrm{4}} \\ $$

Question Number 152912    Answers: 1   Comments: 0

Question Number 152910    Answers: 0   Comments: 0

āˆ«^ x^x^x dx=

$$\int^{} {x}^{{x}^{{x}} } {dx}= \\ $$

Question Number 152907    Answers: 1   Comments: 0

Find a closed form: Ī©=(āˆ«_( 0) ^( 1) ((x^(29) āˆ’x^9 )/(x^(40) +1)) dx)(āˆ«_( 0) ^( 1) ((x^(29) āˆ’2x^9 )/(x^(40) +4))dx)

$$\mathrm{Find}\:\mathrm{a}\:\mathrm{closed}\:\mathrm{form}: \\ $$$$\Omega=\left(\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{29}} āˆ’\mathrm{x}^{\mathrm{9}} }{\mathrm{x}^{\mathrm{40}} +\mathrm{1}}\:\mathrm{dx}\right)\left(\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{x}^{\mathrm{29}} āˆ’\mathrm{2x}^{\mathrm{9}} }{\mathrm{x}^{\mathrm{40}} +\mathrm{4}}\mathrm{dx}\right) \\ $$

Question Number 152901    Answers: 0   Comments: 0

Question Number 152900    Answers: 1   Comments: 0

Question Number 152899    Answers: 0   Comments: 0

Question Number 152898    Answers: 1   Comments: 2

Question Number 153130    Answers: 1   Comments: 0

if x^5 +x^4 +x^3 +2x^2 +x+1=0 find x^3 - (1/x^3 ) = ?

$$\mathrm{if}\:\:\:\mathrm{x}^{\mathrm{5}} +\mathrm{x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} +\mathrm{2x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}=\mathrm{0} \\ $$$$\mathrm{find}\:\:\:\mathrm{x}^{\mathrm{3}} \:-\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{3}} }\:=\:? \\ $$

Question Number 152892    Answers: 0   Comments: 2

Solve for real numbers the following system of equations: { ((x^2 - yz = 3)),((y^2 - xz = 1)),((z^2 - xy = - 1)) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{following} \\ $$$$\mathrm{system}\:\mathrm{of}\:\mathrm{equations}: \\ $$$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:-\:\mathrm{yz}\:=\:\mathrm{3}}\\{\mathrm{y}^{\mathrm{2}} \:-\:\mathrm{xz}\:=\:\mathrm{1}}\\{\mathrm{z}^{\mathrm{2}} \:-\:\mathrm{xy}\:=\:-\:\mathrm{1}}\end{cases} \\ $$

Question Number 152889    Answers: 0   Comments: 7

Ī£_(k=1) ^n ((Ī£_(k=1) ^n k^Ī± )/((n+1)^Ī± Ī£_(k=1) ^n (1+nĪ±)))=(1/(6o)) Ī±=? Ī±=?q

$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\alpha} }{\left({n}+\mathrm{1}\right)^{\alpha} \:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{1}+{n}\alpha\right)}=\frac{\mathrm{1}}{\mathrm{6}{o}}\: \\ $$$$\alpha=? \\ $$$$\alpha=?{q} \\ $$

Question Number 152887    Answers: 1   Comments: 0

āˆ«_1 ^( 2) (3/( (āˆš((x^2 +3)^3 ))))

$$\int_{\mathrm{1}} ^{\:\mathrm{2}} \:\:\frac{\mathrm{3}}{\:\sqrt{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{3}\right)^{\mathrm{3}} }} \\ $$

Question Number 154210    Answers: 0   Comments: 2

Question Number 152881    Answers: 0   Comments: 0

āˆ«_( 0) ^( 1) Li_2 ((x/(1 - x))) log(x) log(1 - x) dx = ?

$$\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{Li}_{\mathrm{2}} \:\left(\frac{\mathrm{x}}{\mathrm{1}\:-\:\mathrm{x}}\right)\:\mathrm{log}\left(\mathrm{x}\right)\:\mathrm{log}\left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 152879    Answers: 0   Comments: 3

lim_(xā†’+oo) ((Ī£_(k=1) ^n k^Ī± )/((n+1)^Ī± Ī£_(k=1) ^n (nĪ±+1)))=(1/(6o)) .Ī±=?

$$\underset{{x}\rightarrow+{oo}} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{k}^{\alpha} \:\:\:}{\left({n}+\mathrm{1}\right)^{\alpha} \:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left({n}\alpha+\mathrm{1}\right)}=\frac{\mathrm{1}}{\mathrm{6}{o}}\:\:.\alpha=? \\ $$

Question Number 152904    Answers: 5   Comments: 0

Question Number 152903    Answers: 5   Comments: 0

Question Number 152874    Answers: 2   Comments: 0

solve: I := āˆ«_0 ^( āˆž) ((( tanh (x) )/x) )^( 2) dx = ? m.n.

$$ \\ $$$$\:\:\:{solve}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{I}\::=\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\:{tanh}\:\left({x}\right)\:}{{x}}\:\right)^{\:\mathrm{2}} {dx}\:=\:? \\ $$$$\:\:{m}.{n}. \\ $$

Question Number 152863    Answers: 1   Comments: 1

āˆ«_(āˆ’āˆž) ^( āˆž) (1/( (āˆš(x^2 +1)))) dx

$$\: \\ $$$$\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\int_{āˆ’\infty} ^{\:\infty} \:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\:\:{dx} \\ $$$$\: \\ $$$$\: \\ $$

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