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Question Number 149349    Answers: 0   Comments: 0

Question Number 149339    Answers: 1   Comments: 0

...nice......mathematics... ln( 2) −Σ_(n=1 ) ^∞ ((ζ ( 2n+1 )−1)/(n + 1)) = ? ....m.n....

$$\:\:\:\:\:\:\:\:\:\:...{nice}......{mathematics}... \\ $$$$\:\:\:\:\:\:{ln}\left(\:\mathrm{2}\right)\:−\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{\zeta\:\left(\:\mathrm{2}{n}+\mathrm{1}\:\right)−\mathrm{1}}{{n}\:+\:\mathrm{1}}\:=\:? \\ $$$$\:\:\:\:....{m}.{n}.... \\ $$

Question Number 149331    Answers: 1   Comments: 2

Question Number 149329    Answers: 1   Comments: 1

Question Number 149323    Answers: 2   Comments: 0

How many ordered pairs (a,b) with b < a < 100 , a,b ∈ N such that (a/b) both ((a+1)/(b+1)) are integers .

$${How}\:\:{many}\:\:{ordered}\:\:{pairs}\:\left({a},{b}\right)\:{with}\:\:{b}\:<\:{a}\:<\:\mathrm{100}\:\:,\:\:{a},{b}\:\in\:\mathbb{N}\:\:{such}\:\:{that}\:\:\frac{{a}}{{b}}\:\:{both}\:\:\:\frac{{a}+\mathrm{1}}{{b}+\mathrm{1}}\:\:{are}\:\:{integers}\:\:.\: \\ $$

Question Number 149322    Answers: 0   Comments: 5

((Σ_(n=1) ^(99) ((√(10 + (√n)))))/(Σ_(n=1) ^(99) ((√(10 - (√n)))))) = ?

$$\frac{\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\left(\sqrt{\mathrm{10}\:+\:\sqrt{\boldsymbol{{n}}}}\right)}{\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\mathrm{99}} {\sum}}\left(\sqrt{\mathrm{10}\:-\:\sqrt{\boldsymbol{{n}}}}\right)}\:=\:? \\ $$

Question Number 149317    Answers: 2   Comments: 0

lim_(x→∞) (√(x^6 +5x^3 ))−x

$${lim}_{{x}\rightarrow\infty} \sqrt{{x}^{\mathrm{6}} +\mathrm{5}{x}^{\mathrm{3}} }−{x} \\ $$

Question Number 149309    Answers: 3   Comments: 0

if t=tanx what is the value of sinx and cosx ?

$${if}\:\:\:\:{t}={tanx}\:{what}\:{is}\:{the}\:{value}\:{of} \\ $$$${sinx}\:{and}\:{cosx}\:? \\ $$

Question Number 149296    Answers: 2   Comments: 0

if a ; b > 0 find [ (((a^2 + 4)(b^2 + 9))/(ab)) ]_(min) = ?

$${if}\:\:\:{a}\:;\:{b}\:>\:\mathrm{0} \\ $$$${find}\:\:\:\left[\:\frac{\left({a}^{\mathrm{2}} \:+\:\mathrm{4}\right)\left({b}^{\mathrm{2}} \:+\:\mathrm{9}\right)}{{ab}}\:\right]_{\boldsymbol{{min}}} =\:? \\ $$

Question Number 149292    Answers: 1   Comments: 0

if sin2𝛂 = - (1/3) find 3tg^2 (((3𝛑)/4) - 𝛂) = ?

$${if}\:\:\:{sin}\mathrm{2}\boldsymbol{\alpha}\:=\:-\:\frac{\mathrm{1}}{\mathrm{3}} \\ $$$${find}\:\:\:\mathrm{3}{tg}^{\mathrm{2}} \left(\frac{\mathrm{3}\boldsymbol{\pi}}{\mathrm{4}}\:-\:\boldsymbol{\alpha}\right)\:=\:? \\ $$

Question Number 149291    Answers: 2   Comments: 0

f(x) = 2sinx^2 −cos^2 x−2 f^′ (x) = ?

$${f}\left({x}\right)\:=\:\mathrm{2}{sinx}^{\mathrm{2}} −{cos}^{\mathrm{2}} {x}−\mathrm{2} \\ $$$${f}\:^{'} \left({x}\right)\:=\:? \\ $$

Question Number 152597    Answers: 0   Comments: 0

Question Number 149275    Answers: 0   Comments: 0

Question Number 149274    Answers: 1   Comments: 0

Question Number 149273    Answers: 3   Comments: 0

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

$$\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{e}^{\mathrm{tan}\:\mathrm{x}} \:\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}{\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}\:\mathrm{dx}\:=? \\ $$

Question Number 149268    Answers: 4   Comments: 0

Question Number 149266    Answers: 1   Comments: 0

Solve for natural numbers: x^2 + y^2 +x + y = 3xy

$${Solve}\:{for}\:{natural}\:{numbers}: \\ $$$${x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+{x}\:+\:{y}\:=\:\mathrm{3}{xy} \\ $$

Question Number 149271    Answers: 2   Comments: 0

Question Number 149259    Answers: 1   Comments: 0

cos^(−1) ((√(sin (41°)sin (19°)+(3/4))) )=?

$$\:\mathrm{cos}^{−\mathrm{1}} \left(\sqrt{\mathrm{sin}\:\left(\mathrm{41}°\right)\mathrm{sin}\:\left(\mathrm{19}°\right)+\frac{\mathrm{3}}{\mathrm{4}}}\:\right)=? \\ $$

Question Number 149252    Answers: 4   Comments: 0

e^y =(sin x)(cos x) find (dy/dx)

$$ \\ $$$${e}^{{y}} =\left(\mathrm{sin}\:{x}\right)\left(\mathrm{cos}\:{x}\right) \\ $$$${find}\:\frac{{dy}}{{dx}} \\ $$

Question Number 149251    Answers: 2   Comments: 0

lim_(x→0^+ ) ((tan x (√(tan x))−sin x (√(sin x)))/(x^3 (√x))) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\mathrm{tan}\:\mathrm{x}\:\sqrt{\mathrm{tan}\:\mathrm{x}}−\mathrm{sin}\:\mathrm{x}\:\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\mathrm{3}} \:\sqrt{\mathrm{x}}}\:=? \\ $$$$ \\ $$

Question Number 149247    Answers: 0   Comments: 3

Question Number 149244    Answers: 0   Comments: 0

Let a,b,c be positive real numbers with sum 3. Prove that (1/a)+(1/b)+(1/c) ≥ (3/(2a^2 +bc))+(3/(2b^2 +ac))+(3/(2c^2 +ab))

$$\:\:{Let}\:{a},{b},{c}\:{be}\:{positive}\:{real}\:{numbers} \\ $$$${with}\:{sum}\:\mathrm{3}.\:{Prove}\:{that}\: \\ $$$$\:\:\frac{\mathrm{1}}{{a}}+\frac{\mathrm{1}}{{b}}+\frac{\mathrm{1}}{{c}}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}{a}^{\mathrm{2}} +{bc}}+\frac{\mathrm{3}}{\mathrm{2}{b}^{\mathrm{2}} +{ac}}+\frac{\mathrm{3}}{\mathrm{2}{c}^{\mathrm{2}} +{ab}} \\ $$

Question Number 149241    Answers: 2   Comments: 0

Question Number 149240    Answers: 1   Comments: 0

solve for real number x^4 −3x^2 +1 =(√((4/(4−x^2 )) −1))

$$\:\:\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{number}\: \\ $$$$\:\mathrm{x}^{\mathrm{4}} −\mathrm{3x}^{\mathrm{2}} +\mathrm{1}\:=\sqrt{\frac{\mathrm{4}}{\mathrm{4}−\mathrm{x}^{\mathrm{2}} }\:−\mathrm{1}} \\ $$$$\:\: \\ $$

Question Number 149239    Answers: 1   Comments: 0

lim_(x→a^+ ) (((√x)−(√a) −(√(x−a)))/( (√(x^2 −a^2 )))) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{a}^{+} } {\mathrm{lim}}\:\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{a}}\:−\sqrt{\mathrm{x}−\mathrm{a}}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} }}\:=?\: \\ $$

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