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Question Number 190702    Answers: 0   Comments: 0

Calcule: I=∫_o ^(𝛑/3) x^2 (sinx)^2 dx

$$\boldsymbol{\mathrm{Calcule}}:\:\:\boldsymbol{\mathrm{I}}=\int_{\boldsymbol{\mathrm{o}}} ^{\frac{\boldsymbol{\pi}}{\mathrm{3}}} \boldsymbol{\mathrm{x}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{sinx}}\right)^{\mathrm{2}} \boldsymbol{\mathrm{dx}} \\ $$

Question Number 190700    Answers: 1   Comments: 0

Question Number 190699    Answers: 0   Comments: 0

Question Number 190694    Answers: 1   Comments: 1

Question Number 190692    Answers: 1   Comments: 0

calculate 𝛗 = ∫_0 ^( ∞) (( sin^( 3) (x ) ln( x ))/x) dx = ? @ nice βˆ’ mathematics

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{calculate}\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{sin}^{\:\mathrm{3}} \left({x}\:\right)\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:\mathrm{d}{x}\:=\:?\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:@\:\mathrm{nice}\:βˆ’\:\mathrm{mathematics}\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 190686    Answers: 2   Comments: 0

find the value of x e^(ln(2x+5)) =lne(8x+3)

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x} \\ $$$${e}^{\mathrm{ln}\left(\mathrm{2}{x}+\mathrm{5}\right)} =\mathrm{ln}{e}\left(\mathrm{8}{x}+\mathrm{3}\right) \\ $$

Question Number 190679    Answers: 2   Comments: 0

lim_(xβ†’βˆž) (e^x /x^(60!) )=? pleas solve this

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{{e}^{{x}} }{{x}^{\mathrm{60}!} }=? \\ $$$${pleas}\:{solve}\:{this} \\ $$

Question Number 190678    Answers: 2   Comments: 0

Question Number 190677    Answers: 0   Comments: 0

If x ∈ R a_1 ,a_2 ,a_3 , b_1 ,b_2 ,b_3 > 0 Then prove that: a_1 ^(sin^2 x) b_1 ^(cos^2 x) + a_2 ^(sin^2 x) b_2 ^(cos^2 x) + a_3 ^(sin^2 x) b_3 ^(cos^2 x) ≀ ≀ (a_1 + a_2 + a_3 )^(sin^2 x) (b_1 + b_2 + b_3 )^(cos^2 x)

$$\mathrm{If}\:\:\mathrm{x}\:\in\:\mathbb{R} \\ $$$$\:\:\:\:\:\mathrm{a}_{\mathrm{1}} ,\mathrm{a}_{\mathrm{2}} ,\mathrm{a}_{\mathrm{3}} \:,\:\mathrm{b}_{\mathrm{1}} ,\mathrm{b}_{\mathrm{2}} ,\mathrm{b}_{\mathrm{3}} \:>\:\mathrm{0} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{a}_{\mathrm{1}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{1}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:+\:\mathrm{a}_{\mathrm{2}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{2}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:+\:\mathrm{a}_{\mathrm{3}} ^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{b}_{\mathrm{3}} ^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\leqslant \\ $$$$\leqslant\:\left(\mathrm{a}_{\mathrm{1}} +\:\mathrm{a}_{\mathrm{2}} +\:\mathrm{a}_{\mathrm{3}} \right)^{\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\left(\mathrm{b}_{\mathrm{1}} +\:\mathrm{b}_{\mathrm{2}} +\:\mathrm{b}_{\mathrm{3}} \right)^{\boldsymbol{\mathrm{cos}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \\ $$

Question Number 190672    Answers: 2   Comments: 0

Question Number 190667    Answers: 2   Comments: 0

Question Number 190665    Answers: 0   Comments: 0

Question Number 190660    Answers: 2   Comments: 0

if y = sin^(βˆ’1) (2x(√(1βˆ’x^2 ))) where x∈[βˆ’(1/( (√2))), (1/( (√2)))], then find (dy/dx).

$${if}\:{y}\:=\:{sin}^{βˆ’\mathrm{1}} \left(\mathrm{2}{x}\sqrt{\mathrm{1}βˆ’{x}^{\mathrm{2}} }\right)\:{where}\:{x}\in\left[βˆ’\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}},\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\right],\:{then}\:{find}\:\frac{{dy}}{{dx}}. \\ $$

Question Number 190659    Answers: 0   Comments: 0

Prove that: ((cos (Ο€/9)))^(1/3) βˆ’ ((cos ((2Ο€)/9)))^(1/3) βˆ’ ((cos ((4Ο€)/9)))^(1/3) = ((3 βˆ’ (3/2) (9)^(1/3) ))^(1/3)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\pi}{\mathrm{9}}}\:βˆ’\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{9}}}\:βˆ’\:\sqrt[{\mathrm{3}}]{\mathrm{cos}\:\frac{\mathrm{4}\pi}{\mathrm{9}}} \\ $$$$=\:\sqrt[{\mathrm{3}}]{\mathrm{3}\:βˆ’\:\frac{\mathrm{3}}{\mathrm{2}}\:\sqrt[{\mathrm{3}}]{\mathrm{9}}}\: \\ $$

Question Number 190652    Answers: 0   Comments: 0

Question Number 190636    Answers: 0   Comments: 0

Question Number 190634    Answers: 2   Comments: 6

The volume of a right circular cone is 5litres. Calculate the volume of the part into which the cone is divided by a plane parallel to the base oneβˆ’third of the way down from the vertex to the base giving your answer to the nearest millimetres. Help!

$$\mathrm{The}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{a}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone} \\ $$$$\mathrm{is}\:\mathrm{5litres}.\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{part}\:\mathrm{into}\:\mathrm{which}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{is} \\ $$$$\mathrm{divided}\:\mathrm{by}\:\mathrm{a}\:\mathrm{plane}\:\mathrm{parallel}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{base}\:\mathrm{one}βˆ’\mathrm{third}\:\mathrm{of}\:\mathrm{the}\:\mathrm{way}\:\mathrm{down} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{to}\:\mathrm{the}\:\mathrm{base}\:\mathrm{giving} \\ $$$$\mathrm{your}\:\mathrm{answer}\:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\: \\ $$$$\mathrm{millimetres}. \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$

Question Number 190629    Answers: 0   Comments: 1

Question Number 190625    Answers: 1   Comments: 0

solve in R : ⌊ (1/x) βŒ‹ + ⌊ x βŒ‹ = 2

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{solve}\:\mathrm{in}\:\:\:\mathbb{R}\:\:\:: \\ $$$$\:\:\:\:\:\:\:\:\lfloor\:\frac{\mathrm{1}}{{x}}\:\rfloor\:\:+\:\lfloor\:{x}\:\rfloor\:=\:\mathrm{2}\:\:\:\:\: \\ $$$$ \\ $$

Question Number 190624    Answers: 2   Comments: 0

Question Number 190623    Answers: 2   Comments: 0

calculate : Ξ£_(n=1) ^∞ (( (βˆ’1)^( nβˆ’1) )/n) cos ((( nΟ€)/3) ) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{calculate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(βˆ’\mathrm{1}\right)^{\:{n}βˆ’\mathrm{1}} }{{n}}\:\mathrm{cos}\:\left(\frac{\:{n}\pi}{\mathrm{3}}\:\right)\:=? \\ $$$$ \\ $$

Question Number 190639    Answers: 1   Comments: 0

The points A, B and C have position vectors iβˆ’j , 5iβˆ’3j and 11iβˆ’6j respectively . Show that A, B and C are collinear.

$$\:{The}\:{points}\:{A},\:{B}\:{and}\:{C}\:{have}\:{position} \\ $$$$\:{vectors}\:{i}βˆ’{j}\:,\:\mathrm{5}{i}βˆ’\mathrm{3}{j}\:{and}\:\mathrm{11}{i}βˆ’\mathrm{6}{j}\: \\ $$$${respectively}\:.\:{Show}\:{that}\:{A},\:{B}\:{and}\:{C} \\ $$$${are}\:{collinear}. \\ $$

Question Number 190622    Answers: 1   Comments: 0

prove : ∫_(βˆ’βˆž) ^( ∞) ((( x)/( cosh(x))))^2 dx= Ξ£_(k=1) ^∞ (1/( k^2 ))

$$ \\ $$$$\:\:\:{prove}\:: \\ $$$$\:\:\int_{βˆ’\infty} ^{\:\infty} \:\:\:\left(\frac{\:{x}}{ {cosh}\left({x}\right)}\right)^{\mathrm{2}} \mathrm{d}{x}=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{k}\:^{\mathrm{2}} }\:\: \: \\ $$$$ \\ $$

Question Number 190618    Answers: 1   Comments: 0

f(xβˆ’f(x))=x ,f(5)=βˆ’1 f(βˆ’1)=?

$$\mathrm{f}\left(\mathrm{x}βˆ’\mathrm{f}\left(\mathrm{x}\right)\right)=\mathrm{x}\:\:,\mathrm{f}\left(\mathrm{5}\right)=βˆ’\mathrm{1} \\ $$$$\mathrm{f}\left(βˆ’\mathrm{1}\right)=? \\ $$

Question Number 190615    Answers: 1   Comments: 1

Question Number 190611    Answers: 0   Comments: 2

Montrer que: 1β€’c^2 =a^2 +b^2 2β€’ rayon r=(c/(1+(√2)))βˆ’((a+b)/(2+(√2)))

$$\mathrm{Montrer}\:\mathrm{que}: \\ $$$$\mathrm{1}\bullet\boldsymbol{\mathrm{c}}^{\mathrm{2}} =\boldsymbol{\mathrm{a}}^{\mathrm{2}} +\boldsymbol{\mathrm{b}}^{\mathrm{2}} \\ $$$$\mathrm{2}\bullet\:\mathrm{rayon}\:\:\:\:\:\boldsymbol{\mathrm{r}}=\frac{\boldsymbol{\mathrm{c}}}{\mathrm{1}+\sqrt{\mathrm{2}}}βˆ’\frac{\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{b}}}{\mathrm{2}+\sqrt{\mathrm{2}}} \\ $$$$ \\ $$

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