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Question Number 117163 Answers: 2 Comments: 0

∫ ((4 dx)/(x (√(x^2 −1)))) =?

$$\:\:\:\int\:\frac{\mathrm{4}\:\mathrm{dx}}{\mathrm{x}\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}}\:=? \\ $$

Question Number 117144 Answers: 1 Comments: 0

S_n =Π_(k=1) ^n cos(kx)=?????? please help

$$\boldsymbol{{S}}_{\boldsymbol{{n}}} =\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{n}}} {\prod}}\boldsymbol{{cos}}\left(\boldsymbol{{kx}}\right)=?????? \\ $$$$\boldsymbol{{please}}\:\boldsymbol{{help}} \\ $$

Question Number 117143 Answers: 1 Comments: 0

Question Number 117142 Answers: 0 Comments: 0

Question Number 117140 Answers: 1 Comments: 0

Question Number 117216 Answers: 1 Comments: 0

∫_0 ^(π/2) (dx/( (√(sin x)))) =?

$$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{\mathrm{dx}}{\:\sqrt{\mathrm{sin}\:\mathrm{x}}}\:=?\: \\ $$$$ \\ $$

Question Number 117213 Answers: 3 Comments: 0

∫_0 ^( ∞) ((tan^(−1) ((x/4))−tan^(−1) ((x/6)))/x) dx =?

$$\:\:\underset{\mathrm{0}} {\overset{\:\:\:\:\:\:\:\infty} {\int}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{4}}\right)−\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{x}}{\mathrm{6}}\right)}{\mathrm{x}}\:\mathrm{dx}\:=? \\ $$

Question Number 117184 Answers: 0 Comments: 0

let I be an interval.prove that c^((∞)) (I)=∩_(n=1) ^∞ c^n (I)

$${let}\:{I}\:{be}\:{an}\:{interval}.{prove}\:{that} \\ $$$${c}^{\left(\infty\right)} \left({I}\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\cap}}{c}^{{n}} \left({I}\right) \\ $$

Question Number 117122 Answers: 0 Comments: 4

Given a,b,c ∈R^3 such that abc=1. Show that: (a−1+(1/b))(b−1+(1/c))(c−1+(1/a))≤1

$$\mathrm{Given}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\in\mathbb{R}^{\mathrm{3}} \:\mathrm{such}\:\mathrm{that}\:\mathrm{abc}=\mathrm{1}.\:\mathrm{Show}\:\mathrm{that}: \\ $$$$\:\:\:\:\:\left(\mathrm{a}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{b}}\right)\left(\mathrm{b}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{c}}\right)\left(\mathrm{c}−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{a}}\right)\leqslant\mathrm{1} \\ $$

Question Number 117116 Answers: 0 Comments: 2

x , y ,z , t ∈ Z. x and y are x are respectively the divisor of y and t. Justify the existence of k ∈ Z such that yt=xzk. Deduct that x^(m ) is a divisor of y^m where m ∈ N.

$${x}\:,\:{y}\:,{z}\:,\:{t}\:\in\:\mathbb{Z}. \\ $$$${x}\:{and}\:{y}\:{are}\:{x}\:{are}\:{respectively}\:{the} \\ $$$${divisor}\:{of}\:{y}\:{and}\:{t}. \\ $$$${Justify}\:{the}\:{existence}\:{of}\:{k}\:\in\:\mathbb{Z}\:{such} \\ $$$${that}\:{yt}={xzk}. \\ $$$${Deduct}\:{that}\:{x}^{{m}\:} {is}\:{a}\:{divisor}\:{of}\:{y}^{{m}} \\ $$$${where}\:{m}\:\in\:\mathbb{N}. \\ $$

Question Number 117107 Answers: 1 Comments: 0

Question Number 117101 Answers: 2 Comments: 0

If sin^2 θ and cos^2 θ are the roots of quadratic equation, find the equation.

$$\mathrm{If}\:\mathrm{sin}^{\mathrm{2}} \theta\:\mathrm{and}\:\mathrm{cos}^{\mathrm{2}} \theta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\mathrm{quadratic}\:\mathrm{equation},\:\mathrm{find}\:\mathrm{the}\:\mathrm{equation}. \\ $$

Question Number 117100 Answers: 3 Comments: 0

∫ sin^6 (2x) cos^6 (2x) dx =?

$$\int\:\mathrm{sin}\:^{\mathrm{6}} \left(\mathrm{2x}\right)\:\mathrm{cos}\:^{\mathrm{6}} \left(\mathrm{2x}\right)\:\mathrm{dx}\:=? \\ $$

Question Number 117148 Answers: 2 Comments: 0

∫ x^6 e^(−4x^2 ) dx =?

$$\:\:\:\int\:\mathrm{x}^{\mathrm{6}} \:\mathrm{e}^{−\mathrm{4x}^{\mathrm{2}} } \:\mathrm{dx}\:=? \\ $$

Question Number 117147 Answers: 2 Comments: 0

lim_(x→0) (((cosh (2x))/(cosh (x))))^(1/x^2 ) =?

$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{cosh}\:\left(\mathrm{2x}\right)}{\mathrm{cosh}\:\left(\mathrm{x}\right)}\right)^{\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }} \:=? \\ $$

Question Number 117095 Answers: 1 Comments: 0

Question Number 117090 Answers: 0 Comments: 0

Question Number 117089 Answers: 0 Comments: 0

Question Number 117088 Answers: 2 Comments: 0

... advanced calculus... evsluate :: I=∫_0 ^( ∞) ((xsin(2x))/(x^2 +4)) dx =? hint: Φ=∫_0 ^( ∞) ((cos(px))/(x^2 +1))dx =(π/2)e^(−p) (p>0) .m.n 1970

$$\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:\:\:{calculus}... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{evsluate}\::: \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\mathrm{I}=\int_{\mathrm{0}} ^{\:\infty} \frac{{xsin}\left(\mathrm{2}{x}\right)}{{x}^{\mathrm{2}} +\mathrm{4}}\:{dx}\:=? \\ $$$$\:\:\:\:\:{hint}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left({px}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\:=\frac{\pi}{\mathrm{2}}{e}^{−{p}} \:\:\:\:\left({p}>\mathrm{0}\right)\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:.{m}.{n}\:\mathrm{1970} \\ $$$$ \\ $$

Question Number 117087 Answers: 2 Comments: 0

... nice calculus... please evaluate :: Ω=∫_0 ^( 1) ((x^4 +1)/(x^6 +1))dx =??? m.n.1970

$$\:\:\:\:\:\:\:\:\:...\:\:{nice}\:\:{calculus}... \\ $$$$\:{please}\:\:{evaluate}\::: \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{\mathrm{4}} +\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx}\:=??? \\ $$$$\:\:\:\:\:\:\:\:\:{m}.{n}.\mathrm{1970} \\ $$

Question Number 117086 Answers: 2 Comments: 0

...nice mathematics... evaluate ... lim_(n→∞) (Π_(k=1) ^n sin(((kπ)/(4n))))^(1/n) =? ...m.n.1970...

$$\:\:\:\:\:\:\:...{nice}\:\:{mathematics}... \\ $$$$\:{evaluate}\:... \\ $$$$\:\:\:\:\:\:\:{lim}_{{n}\rightarrow\infty} \left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\prod}}{sin}\left(\frac{{k}\pi}{\mathrm{4}{n}}\right)\right)^{\frac{\mathrm{1}}{{n}}} =? \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{m}.{n}.\mathrm{1970}... \\ $$$$ \\ $$

Question Number 117085 Answers: 6 Comments: 0

Question Number 117082 Answers: 1 Comments: 0

If 6−3x is the geometric mean between the integer x^2 +2 and 2^ what are the values of x ?

$$\mathrm{If}\:\mathrm{6}−\mathrm{3x}\:\mathrm{is}\:\mathrm{the}\:\mathrm{geometric}\:\mathrm{mean}\:\mathrm{between} \\ $$$$\mathrm{the}\:\mathrm{integer}\:\mathrm{x}^{\mathrm{2}} +\mathrm{2}\:\mathrm{and}\:\bar {\mathrm{2}}\:\mathrm{what}\:\mathrm{are}\:\mathrm{the}\: \\ $$$$\mathrm{values}\:\mathrm{of}\:\mathrm{x}\:? \\ $$

Question Number 117078 Answers: 1 Comments: 0

lim_(n→∞) [ tan ((π/(2n))).tan (((2π)/(2n))).tan (((3π)/(2n)))...tan (((nπ)/(2n)))]^(1/n) =?

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

Question Number 117066 Answers: 1 Comments: 0

Given f(x)= ((x−1)/(x+1)) . If f^2 (x)=f(f(x)), f^3 (x)=f(f(f(x))) , f^(1998) (x) = g(x) then ∫_(1/e) ^1 g(x) dx = _?

Question Number 117061 Answers: 2 Comments: 3

How many 5-digits positive integers x_1 x_2 x_3 x_4 x_5 are there such that x_1 ≤x_2 ≤x_3 ≤x_4 ≤x_5 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{5}-\mathrm{digits}\:\mathrm{positive}\:\mathrm{integers} \\ $$$${x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} {x}_{\mathrm{5}} \:\mathrm{are}\:\mathrm{there}\:\mathrm{such}\:\mathrm{that}\: \\ $$$${x}_{\mathrm{1}} \leqslant{x}_{\mathrm{2}} \leqslant{x}_{\mathrm{3}} \leqslant{x}_{\mathrm{4}} \leqslant{x}_{\mathrm{5}} ? \\ $$

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