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Question Number 41855 Answers: 0 Comments: 1

lim_(x→+∞ ) ((n!)/(ln (1+n!))) find the limit

$$\underset{{x}\rightarrow+\infty\:\:} {\mathrm{lim}}\frac{\mathrm{n}!}{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{n}!\right)} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{limit} \\ $$

Question Number 41854 Answers: 0 Comments: 0

1)calculate ∫_0 ^1 ln(1+x+x^2 +x^3 )dx 2)then find the value of ∫_0 ^1 ln( 1−x^5 ) dx 3) find the value of Σ_(n=1) ^∞ (1/(n(5n+1))) .

$$\left.\mathrm{1}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \:+{x}^{\mathrm{3}} \right){dx} \\ $$$$\left.\mathrm{2}\right){then}\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{ln}\left(\:\mathrm{1}−{x}^{\mathrm{5}} \right)\:{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\mathrm{1}}{{n}\left(\mathrm{5}{n}+\mathrm{1}\right)}\:. \\ $$$$ \\ $$$$ \\ $$

Question Number 41851 Answers: 0 Comments: 2

Question Number 41847 Answers: 1 Comments: 0

let f(x) = ∫_0 ^(π/4) (dt/(x +tan(t))) 1) find anoher expression off (x) 2) calculate ∫_0 ^(π/4) (dt/(2+tan(t))) and A(θ) = ∫_0 ^(π/4) (dt/(sinθ+tant)) 3) calculate ∫_0 ^(π/4) (dt/((1+tant)^2 ))

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\:\frac{{dt}}{{x}\:+{tan}\left({t}\right)} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{anoher}\:{expression}\:{off}\:\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{dt}}{\mathrm{2}+{tan}\left({t}\right)}\:\:\:{and}\:\:{A}\left(\theta\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{{sin}\theta+{tant}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\:\frac{{dt}}{\left(\mathrm{1}+{tant}\right)^{\mathrm{2}} } \\ $$

Question Number 41846 Answers: 1 Comments: 0

find ∫ (dx/((√(1+x^2 )) +(√(1−x^2 ))))

$${find}\:\:\int\:\:\:\:\:\frac{{dx}}{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}\:\:+\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} \\ $$

Question Number 41845 Answers: 1 Comments: 0

1)find ∫ (x/((√(1+x)) +(√(1−x)))) dx 2) calculate ∫_1 ^3 (x/((√(1+x)) +(√(1−x)))) dx

$$\left.\mathrm{1}\right){find}\:\:\:\:\int\:\:\:\:\:\:\:\:\:\frac{{x}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\:\:\frac{{x}}{\sqrt{\mathrm{1}+{x}}\:+\sqrt{\mathrm{1}−{x}}}\:{dx} \\ $$

Question Number 41848 Answers: 0 Comments: 3

let f(a) = ∫_0 ^(π/2) (dx/(1+asinx)) with a∈R 1) find a simple form of f(a) 2) calculate ∫_0 ^(π/2) (dx/(1+sinx)) and ∫_0 ^(π/2) (dx/(1+2sinx)) 3) find the value of ∫_0 ^(π/2) ((cosx)/((1+asinx)^2 ))dx 4) find the value of ∫_0 ^(π/2) ((cosx)/((1+sinx)^2 ))dx and ∫_0 ^(π/2) ((cosx)/((1+2sinx)^2 ))dx

$${let}\:{f}\left({a}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\frac{{dx}}{\mathrm{1}+{asinx}}\:\:\:{with}\:{a}\in{R} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({a}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{dx}}{\mathrm{1}+{sinx}}\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{dx}}{\mathrm{1}+\mathrm{2}{sinx}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cosx}}{\left(\mathrm{1}+{asinx}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{4}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cosx}}{\left(\mathrm{1}+{sinx}\right)^{\mathrm{2}} }{dx}\:{and}\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\frac{{cosx}}{\left(\mathrm{1}+\mathrm{2}{sinx}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 42310 Answers: 1 Comments: 1

Question Number 41828 Answers: 1 Comments: 0

{ (((y^y /x)+(x^x /y)=129)),((x^y +y^x =32)) :} find x and y? by k.khaled

$$\begin{cases}{\frac{{y}^{{y}} }{{x}}+\frac{{x}^{{x}} }{{y}}=\mathrm{129}}\\{{x}^{{y}} +{y}^{{x}} =\mathrm{32}}\end{cases} \\ $$$${find}\:{x}\:{and}\:{y}? \\ $$$$\:{by}\:{k}.{khaled} \\ $$

Question Number 41824 Answers: 3 Comments: 0

if ^n C_5 = ^n C_(11) find ^(18) C_n

$$\mathrm{if}\:\:\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{5}} \:=\:\overset{\mathrm{n}} {\:}\mathrm{C}_{\mathrm{11}} \:\:\:\mathrm{find}\:\:\:\:\:\:\overset{\mathrm{18}} {\:}\mathrm{C}_{\mathrm{n}} \\ $$

Question Number 41820 Answers: 1 Comments: 0

If α, β are the roots of the equation ax^2 +bx+c=0, then the value of the determinant determinant ((1,(cos (β−α)),(cos α)),((cos (α−β)),1,(cos β)),((cos α),(cos β),1)) is

$$\mathrm{If}\:\alpha,\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$${ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0},\:\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{determinant} \\ $$$$\begin{vmatrix}{\mathrm{1}}&{\mathrm{cos}\:\left(\beta−\alpha\right)}&{\mathrm{cos}\:\alpha}\\{\mathrm{cos}\:\left(\alpha−\beta\right)}&{\mathrm{1}}&{\mathrm{cos}\:\beta}\\{\mathrm{cos}\:\alpha}&{\mathrm{cos}\:\beta}&{\mathrm{1}}\end{vmatrix}\:\mathrm{is} \\ $$

Question Number 41813 Answers: 1 Comments: 0

Any workings for this simultaneous equation ? x^x + y^y = 31 ........ equation (i) x + y = 5 ........ equation (ii)

$$\mathrm{Any}\:\mathrm{workings}\:\mathrm{for}\:\mathrm{this}\:\mathrm{simultaneous}\:\mathrm{equation}\:? \\ $$$$\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{x}} \:+\:\mathrm{y}^{\mathrm{y}} \:=\:\mathrm{31}\:\:........\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{x}\:+\:\mathrm{y}\:=\:\mathrm{5}\:\:\:\:\:\:........\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 41806 Answers: 1 Comments: 1

Question Number 41798 Answers: 0 Comments: 5

Solve: x^3 − 3x + 4 = 0 how can i know when to use: let x = y + (1/y)

$$\mathrm{Solve}:\:\:\:\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{3x}\:+\:\mathrm{4}\:=\:\mathrm{0} \\ $$$$\mathrm{how}\:\mathrm{can}\:\mathrm{i}\:\mathrm{know}\:\mathrm{when}\:\mathrm{to}\:\mathrm{use}:\:\:\:\:\mathrm{let}\:\mathrm{x}\:=\:\mathrm{y}\:+\:\frac{\mathrm{1}}{\mathrm{y}} \\ $$

Question Number 41797 Answers: 1 Comments: 0

Find x: 48log_a 4 + 5log_4 a = (a/8)

$$\mathrm{Find}\:\mathrm{x}:\:\:\:\:\:\:\:\:\mathrm{48log}_{\mathrm{a}} \mathrm{4}\:\:+\:\:\mathrm{5log}_{\mathrm{4}} \mathrm{a}\:\:=\:\:\frac{\mathrm{a}}{\mathrm{8}} \\ $$

Question Number 41796 Answers: 2 Comments: 0

Find the value of a log_(a + 2) ^( 8) + log_(16) ^( a) = (7/4)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{a} \\ $$$$\:\:\:\:\:\:\mathrm{log}_{\mathrm{a}\:+\:\mathrm{2}} ^{\:\mathrm{8}} \:\:+\:\:\:\:\mathrm{log}_{\mathrm{16}} ^{\:\mathrm{a}} \:\:=\:\:\frac{\mathrm{7}}{\mathrm{4}} \\ $$

Question Number 41780 Answers: 2 Comments: 0

Question Number 41766 Answers: 2 Comments: 0

Question Number 41765 Answers: 2 Comments: 0

find range of the function f defined by f(x)=(1/(1−x^2 ))

$${find}\:{range}\:{of}\:{the}\:{function}\:{f}\: \\ $$$${defined}\:{by}\:{f}\left({x}\right)=\frac{\mathrm{1}}{\mathrm{1}−{x}^{\mathrm{2}} } \\ $$

Question Number 41762 Answers: 0 Comments: 3

let f(x) = ∫_0 ^1 ((ln(1+xt^2 ))/(1+t^2 )) dt 1) find a simple form of f(x) 2) calculate ∫_0 ^1 ((ln(1+t^2 ))/(1+t^2 ))dt 3) calculate ∫_0 ^1 ((ln(1+2t^2 ))/(1+t^2 )) dt

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{ln}\left(\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt} \\ $$

Question Number 41761 Answers: 1 Comments: 0

calculate lim_(x→0) x ln(1−e^(sinx) )

$${calculate}\:{lim}_{{x}\rightarrow\mathrm{0}} \:\:{x}\:{ln}\left(\mathrm{1}−{e}^{{sinx}} \right)\: \\ $$

Question Number 41757 Answers: 0 Comments: 0

Find the fourier sine transform of (1/(x(x^r + a^r )))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{fourier}\:\mathrm{sine}\:\mathrm{transform}\:\mathrm{of}\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{x}\left(\mathrm{x}^{\mathrm{r}} \:+\:\mathrm{a}^{\mathrm{r}} \right)} \\ $$

Question Number 41744 Answers: 1 Comments: 0

Question Number 41754 Answers: 3 Comments: 0

Question Number 41753 Answers: 2 Comments: 0

How far must a man stand in front of a concave mirror of radius 120cm in order to see an erect image of his face four times its actual size?

$${How}\:{far}\:{must}\:{a}\:{man}\:{stand}\:{in}\:{front} \\ $$$${of}\:{a}\:{concave}\:{mirror}\:{of}\:{radius}\:\mathrm{120}{cm} \\ $$$${in}\:{order}\:{to}\:{see}\:{an}\:{erect}\:{image}\:{of} \\ $$$${his}\:{face}\:{four}\:{times}\:{its}\:{actual}\:{size}? \\ $$

Question Number 41726 Answers: 0 Comments: 0

Tank T_1 and T_2 initially contains 100gal of water each.In T_1 the water is pure,whereas 150lb of fertilizer are dissolved in T_2 .By circulating liquid at a rate of 1gal/min and stirring the amount of fertilizer y_1 (t) in T_1 and y_2 (t) in T_2 change with time t. a)with a schematic diagram write the model linear equations for the mixing problem. (b)determine the eigenvalues and⊛ and eigenvectors of the derived equation.

$${Tank}\:{T}_{\mathrm{1}} \:{and}\:{T}_{\mathrm{2}} \:{initially}\:{contains} \\ $$$$\mathrm{100}{gal}\:{of}\:{water}\:{each}.{In}\:{T}_{\mathrm{1}} \:{the} \\ $$$${water}\:{is}\:{pure},{whereas}\:\mathrm{150}{lb}\:{of} \\ $$$${fertilizer}\:{are}\:{dissolved}\:{in}\:{T}_{\mathrm{2}} .{By} \\ $$$${circulating}\:{liquid}\:{at}\:{a}\:{rate}\:{of} \\ $$$$\mathrm{1}{gal}/{min}\:{and}\:{stirring}\:{the}\:{amount} \\ $$$${of}\:{fertilizer}\:{y}_{\mathrm{1}} \left({t}\right)\:{in}\:{T}_{\mathrm{1}} \:{and}\:{y}_{\mathrm{2}} \left({t}\right) \\ $$$${in}\:{T}_{\mathrm{2}} \:{change}\:{with}\:{time}\:{t}. \\ $$$$\left.{a}\right){with}\:{a}\:{schematic}\:{diagram}\:{write} \\ $$$${the}\:{model}\:{linear}\:{equations}\:{for} \\ $$$${the}\:{mixing}\:{problem}. \\ $$$$\left({b}\right){determine}\:{the}\:{eigenvalues}\:{and}\circledast \\ $$$${and}\:{eigenvectors}\:{of}\:{the}\:{derived} \\ $$$${equation}. \\ $$

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