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Question Number 118156 Answers: 0 Comments: 2

(1−x^2 )y′′−8xy′−12y=0

$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}''−\mathrm{8}{xy}'−\mathrm{12}{y}=\mathrm{0} \\ $$

Question Number 118150 Answers: 2 Comments: 1

Determine all function f:R╲{0,1} →R satisfying the functional relation f(x)+f((1/(1−x))) = ((2(1−2x))/(x(1−x))); for x≠0 and x≠1

$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{function}\:\mathrm{f}:\mathbb{R}\diagdown\left\{\mathrm{0},\mathrm{1}\right\}\:\rightarrow\mathbb{R} \\ $$$$\mathrm{satisfying}\:\mathrm{the}\:\mathrm{functional}\:\mathrm{relation} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)+\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\frac{\mathrm{2}\left(\mathrm{1}−\mathrm{2x}\right)}{\mathrm{x}\left(\mathrm{1}−\mathrm{x}\right)};\:\mathrm{for}\:\mathrm{x}\neq\mathrm{0}\:\mathrm{and}\:\mathrm{x}\neq\mathrm{1} \\ $$

Question Number 118145 Answers: 3 Comments: 1

∫ (dx/((x+1)^2 (x^2 +1))) ?

$$\int\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:? \\ $$

Question Number 118142 Answers: 0 Comments: 0

Question Number 118138 Answers: 2 Comments: 0

Question Number 118140 Answers: 2 Comments: 0

lim_(x→0) (([ x^2 ])/(2x)) =?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left[\:{x}^{\mathrm{2}} \:\right]}{\mathrm{2}{x}}\:=?\: \\ $$

Question Number 118133 Answers: 1 Comments: 5

2. Turunan pertama dari f(x)=5x^4 +x(√x)+(6/( (√×)))−sin x−2cos x+5 adalah... a. f^1 (x)=20x^3 +(3/2)(√x)+(6/(×(√×)))+cos x−2sin x b. f^1 (x)=20x^3 +(3/2)(√x)−(3/(x(√x)))−cos x+2sin x c. f^1 (x)=20x^3 +(3/2)(√x)−(1/(x(√x)))−cos x+2sin x d. f^1 (x)=20x^3 +(2/3)(√x)−(3/(x(√x)))+cos x+2sin x e. f^1 (x)=20x^3 +(3/2)(√x)−(1/(x(√x)))+cos x+2sin x

$$\mathrm{2}.\:\mathrm{Turunan}\:\mathrm{pertama}\:\mathrm{dari}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{5x}^{\mathrm{4}} +\mathrm{x}\sqrt{\mathrm{x}}+\frac{\mathrm{6}}{\:\sqrt{×}}−\mathrm{sin}\:\mathrm{x}−\mathrm{2cos}\:\mathrm{x}+\mathrm{5}\:{adalah}... \\ $$$$\mathrm{a}.\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{20x}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{2}}\sqrt{\mathrm{x}}+\frac{\mathrm{6}}{×\sqrt{×}}+\mathrm{cos}\:\mathrm{x}−\mathrm{2sin}\:\mathrm{x} \\ $$$$\mathrm{b}.\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{20x}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{2}}\sqrt{\mathrm{x}}−\frac{\mathrm{3}}{\mathrm{x}\sqrt{\mathrm{x}}}−\mathrm{cos}\:\mathrm{x}+\mathrm{2sin}\:\mathrm{x} \\ $$$$\mathrm{c}.\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{20x}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{2}}\sqrt{\mathrm{x}}−\frac{\mathrm{1}}{\mathrm{x}\sqrt{\mathrm{x}}}−\mathrm{cos}\:\mathrm{x}+\mathrm{2sin}\:\mathrm{x} \\ $$$$\mathrm{d}.\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{20x}^{\mathrm{3}} +\frac{\mathrm{2}}{\mathrm{3}}\sqrt{\mathrm{x}}−\frac{\mathrm{3}}{\mathrm{x}\sqrt{\mathrm{x}}}+\mathrm{cos}\:\mathrm{x}+\mathrm{2sin}\:\mathrm{x} \\ $$$$\mathrm{e}.\:\mathrm{f}^{\mathrm{1}} \left(\mathrm{x}\right)=\mathrm{20x}^{\mathrm{3}} +\frac{\mathrm{3}}{\mathrm{2}}\sqrt{\mathrm{x}}−\frac{\mathrm{1}}{\mathrm{x}\sqrt{\mathrm{x}}}+\mathrm{cos}\:\mathrm{x}+\mathrm{2sin}\:\mathrm{x} \\ $$

Question Number 118120 Answers: 1 Comments: 1

tan (tan x) + tan (2x)=tan (3x)

$$\mathrm{tan}\:\left(\mathrm{tan}\:{x}\right)\:+\:\mathrm{tan}\:\left(\mathrm{2}{x}\right)=\mathrm{tan}\:\left(\mathrm{3}{x}\right) \\ $$

Question Number 118118 Answers: 3 Comments: 0

f(x+y) = f(x) f(y) ∀x ∈ R f(1) = 8 f((2/3)) = ?

$${f}\left({x}+{y}\right)\:=\:{f}\left({x}\right)\:{f}\left({y}\right)\:\:\:\forall{x}\:\in\:\mathbb{R} \\ $$$${f}\left(\mathrm{1}\right)\:=\:\mathrm{8} \\ $$$${f}\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:=\:? \\ $$

Question Number 118113 Answers: 3 Comments: 0

∫ (dx/(x^3 (√(x^2 −a^2 )))) =?

$$\int\:\frac{{dx}}{{x}^{\mathrm{3}} \:\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }}\:=?\: \\ $$

Question Number 118111 Answers: 2 Comments: 0

∫_0 ^(π/4) (x^2 /((x sin x+cos x)^2 )) dx =?

$$\underset{\mathrm{0}} {\overset{\pi/\mathrm{4}} {\int}}\:\frac{\mathrm{x}^{\mathrm{2}} }{\left(\mathrm{x}\:\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{dx}\:=? \\ $$

Question Number 118109 Answers: 0 Comments: 0

Question Number 118104 Answers: 0 Comments: 1

1.) A right circular cone is circumscribed about a sphere of radius(r). If d is the distance from the center of the sphere to the vertex of the cone, show that the volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))). 2.) Find the vertical angle of the cone when it′s volume is minimum.

$$\left.\mathrm{1}.\right) \\ $$$$\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{2}.\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$

Question Number 118090 Answers: 4 Comments: 1

find determinant (((((a^2 +b^2 )/c) c c)),(( a ((b^2 +c^2 )/a) a)),(( b b ((a^2 +c^2 )/b))))=?

$$\mathrm{find}\:\begin{vmatrix}{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} }{\mathrm{c}}\:\:\:\:\:\:\:\:\mathrm{c}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{c}}\\{\:\:\:\:\:\mathrm{a}\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} }{\mathrm{a}}\:\:\:\:\:\:\:\mathrm{a}}\\{\:\:\:\:\:\mathrm{b}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{b}\:\:\:\:\:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} }{\mathrm{b}}}\end{vmatrix}=?\: \\ $$

Question Number 118088 Answers: 1 Comments: 0

lim_(x→∞) ((x.(√(x^2 +1)) .((x^3 +1))^(1/(3 )) )/((2x+1)^3 )) =?

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{x}.\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{1}}\:.\sqrt[{\mathrm{3}\:}]{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}}{\left(\mathrm{2x}+\mathrm{1}\right)^{\mathrm{3}} }\:=? \\ $$

Question Number 118084 Answers: 0 Comments: 1

suppose f: [1,3 ]→ [−1,1 ] such that ∫_1 ^3 f(x) dx = 0 . What the maximum value of ∫_1 ^3 x^(−1) .f(x) dx ?

$$\mathrm{suppose}\:\mathrm{f}:\:\left[\mathrm{1},\mathrm{3}\:\right]\rightarrow\:\left[−\mathrm{1},\mathrm{1}\:\right]\:\mathrm{such} \\ $$$$\mathrm{that}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{0}\:.\:\mathrm{What}\:\mathrm{the}\:\mathrm{maximum} \\ $$$$\mathrm{value}\:\mathrm{of}\:\underset{\mathrm{1}} {\overset{\mathrm{3}} {\int}}\:\mathrm{x}^{−\mathrm{1}} .\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 118083 Answers: 1 Comments: 0

Question Number 118081 Answers: 1 Comments: 3

∫ arc cos (((cos x)/(1+2cos x))) dx =?

$$\:\int\:\mathrm{arc}\:\mathrm{cos}\:\left(\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{x}}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 118078 Answers: 2 Comments: 0

3(cos x+sin x)^4 +6(sin x−cos x)^2 − 3(sin^6 x+cos^6 x) = ?

$$\mathrm{3}\left(\mathrm{cos}\:\mathrm{x}+\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{4}} +\mathrm{6}\left(\mathrm{sin}\:\mathrm{x}−\mathrm{cos}\:\mathrm{x}\right)^{\mathrm{2}} − \\ $$$$\mathrm{3}\left(\mathrm{sin}\:^{\mathrm{6}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{6}} \mathrm{x}\right)\:=\:?\: \\ $$

Question Number 118073 Answers: 2 Comments: 0

Evaluate ∫_0 ^a b(√(1−(x^2 /a^2 ))) dx

$$\mathrm{Evaluate}\:\underset{\mathrm{0}} {\overset{{a}} {\int}}\:{b}\sqrt{\mathrm{1}−\frac{\mathrm{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }}\:{dx}\: \\ $$

Question Number 118070 Answers: 0 Comments: 4

Question Number 118069 Answers: 1 Comments: 0

1.)i) A right circular cone is circumscribed about a sphere of radius(r). If d is the distance from the center of the sphere to the vertex of the cone, show that the volume of the cone,V=((𝛑r^2 (r+d)^2 )/(3(d−r))). ii) Find the vertical angle of the cone when it′s volume is minimum.

$$\left.\mathrm{1}\left..\right)\mathrm{i}\right)\:\mathrm{A}\:\mathrm{right}\:\mathrm{circular}\:\mathrm{cone}\:\mathrm{is}\:\mathrm{circumscribed} \\ $$$$\mathrm{about}\:\mathrm{a}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{radius}\left(\boldsymbol{\mathrm{r}}\right).\:\:\mathrm{If}\:\boldsymbol{\mathrm{d}}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sphere} \\ $$$$\mathrm{to}\:\mathrm{the}\:\mathrm{vertex}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\:\mathrm{show}\:\mathrm{that}\:\mathrm{the} \\ $$$$\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone},\boldsymbol{\mathrm{V}}=\frac{\boldsymbol{\pi\mathrm{r}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{r}}+\boldsymbol{\mathrm{d}}\right)^{\mathrm{2}} }{\mathrm{3}\left(\boldsymbol{\mathrm{d}}−\boldsymbol{\mathrm{r}}\right)}. \\ $$$$\left.\mathrm{ii}\right) \\ $$$$\boldsymbol{\mathrm{F}}\mathrm{ind}\:\mathrm{the}\:\mathrm{vertical}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cone}\:\mathrm{when} \\ $$$$\mathrm{it}'\mathrm{s}\:\mathrm{volume}\:\mathrm{is}\:\mathrm{minimum}. \\ $$

Question Number 118094 Answers: 1 Comments: 0

∫ (dx/(sec x+2)) =?

$$\int\:\frac{\mathrm{dx}}{\mathrm{sec}\:\mathrm{x}+\mathrm{2}}\:=? \\ $$

Question Number 118065 Answers: 1 Comments: 0

Question Number 118064 Answers: 1 Comments: 0

Question Number 118062 Answers: 0 Comments: 2

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