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Question Number 161159 Answers: 2 Comments: 0

Question Number 161150 Answers: 0 Comments: 1

Question Number 161156 Answers: 1 Comments: 0

Calculate lim_(x→+∞) (ln (1+e^(−x) ))^(1/x) lim_(x→0) ((x/(2+sin (1/x)))) lim_(x→0) (((a^x +b^x )/2))^(1/x)

$${Calculate} \\ $$$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\left(\mathrm{ln}\:\left(\mathrm{1}+{e}^{−{x}} \right)\right)^{\frac{\mathrm{1}}{{x}}} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{x}}{\mathrm{2}+\mathrm{sin}\:\frac{\mathrm{1}}{{x}}}\right) \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{{a}^{{x}} +{b}^{{x}} }{\mathrm{2}}\right)^{\frac{\mathrm{1}}{{x}}} \\ $$

Question Number 161146 Answers: 0 Comments: 0

Question Number 161147 Answers: 0 Comments: 0

Question Number 161139 Answers: 0 Comments: 0

Question Number 161136 Answers: 2 Comments: 0

≺ X , τ ≻ is a topological space and A ⊆ X , A^− =^? ∩_(F⊃A) F ( F is closed set )

$$ \\ $$$$\:\:\:\:\:\:\:\prec\:\mathrm{X}\:,\:\tau\:\succ\:{is}\:{a}\:{topological}\:{space} \\ $$$$\:\:\:\:\:\:{and}\:\:\:\mathrm{A}\:\subseteq\:\mathrm{X}\:, \\ $$$$\:\:\:\:\:\:\:\overset{−} {\mathrm{A}}\overset{?} {=}\underset{\mathrm{F}\supset\mathrm{A}} {\cap}\mathrm{F}\:\:\:\:\:\left(\:\mathrm{F}\:{is}\:{closed}\:{set}\:\right) \\ $$$$ \\ $$

Question Number 161133 Answers: 1 Comments: 0

lim_(n→∞) (1/n)∫_0 ^1 (dx/(x(x+(1/n))))=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{n}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{dx}}{\mathrm{x}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{n}}\right)}=? \\ $$

Question Number 161123 Answers: 0 Comments: 0

Find: 𝛀 =lim_(n→∞) (H_n /(n(H_(2n-1) - 2 H_(n-1) )))

$$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\:\frac{\mathrm{H}_{\boldsymbol{\mathrm{n}}} }{\boldsymbol{\mathrm{n}}\left(\mathrm{H}_{\mathrm{2}\boldsymbol{\mathrm{n}}-\mathrm{1}} \:-\:\mathrm{2}\:\mathrm{H}_{\boldsymbol{\mathrm{n}}-\mathrm{1}} \right)} \\ $$

Question Number 161130 Answers: 1 Comments: 0

Given P(x) is polynomial such that P(3x)= P ′(x).P ′′(x) . Find the tangent of curve y = P(x) parallel to the line y= 4x−2.

$$\:{Given}\:{P}\left({x}\right)\:{is}\:{polynomial}\:{such}\:{that} \\ $$$$\:{P}\left(\mathrm{3}{x}\right)=\:{P}\:'\left({x}\right).{P}\:''\left({x}\right)\:.\:{Find}\:{the}\:{tangent} \\ $$$$\:{of}\:{curve}\:{y}\:=\:{P}\left({x}\right)\:{parallel}\:{to}\:{the}\:{line} \\ $$$$\:{y}=\:\mathrm{4}{x}−\mathrm{2}.\: \\ $$

Question Number 161126 Answers: 1 Comments: 2

Question Number 161111 Answers: 1 Comments: 1

lim_(x→0) ((x^2 +2cos x−2)/x^4 ) = (1/a) a=?

$$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} +\mathrm{2cos}\:{x}−\mathrm{2}}{{x}^{\mathrm{4}} }\:=\:\frac{\mathrm{1}}{{a}} \\ $$$$\:\:\:\:{a}=? \\ $$

Question Number 161105 Answers: 1 Comments: 1

Question Number 161102 Answers: 1 Comments: 1

Question Number 161101 Answers: 1 Comments: 0

solve: ∫((x+1)/(x^2 −7x−3))dx

$${solve}: \\ $$$$\:\:\:\int\frac{{x}+\mathrm{1}}{{x}^{\mathrm{2}} −\mathrm{7}{x}−\mathrm{3}}{dx} \\ $$

Question Number 161100 Answers: 0 Comments: 0

f(x^2 )= 2+∫_( 0) ^( x^2 ) f(y) (1−tan y)dy , ∀x∈R f(−π)=?

$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}−\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(−\pi\right)=? \\ $$

Question Number 161096 Answers: 0 Comments: 0

if x;y;z>0 and a;b;c>0 different in pairs and n;k∈N^∗ ((log x^n )/(b^k - c^k )) = ((log y^n )/(c^k - a^k )) = ((log z^n )/(a^k - b^k )) then find (√(xyz))

Question Number 161091 Answers: 1 Comments: 0

Solve for real numbers: (√(1 - x)) = 2x^2 - 1 - 2x (√(1 - x^2 ))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\sqrt{\mathrm{1}\:-\:\mathrm{x}}\:=\:\mathrm{2x}^{\mathrm{2}} \:-\:\mathrm{1}\:-\:\mathrm{2x}\:\sqrt{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} } \\ $$$$ \\ $$

Question Number 161114 Answers: 0 Comments: 0

Let f(x)= sin^3 (2x) for −(π/4)≤x≤(π/4) then Df^(−1) ((1/8))=(a/(b(√b))) so { ((a=?)),((b=?)) :}

$$\:\:{Let}\:{f}\left({x}\right)=\:\mathrm{sin}\:^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:{for}\:−\frac{\pi}{\mathrm{4}}\leqslant{x}\leqslant\frac{\pi}{\mathrm{4}} \\ $$$$\:{then}\:{Df}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{8}}\right)=\frac{{a}}{{b}\sqrt{{b}}}\:{so}\:\begin{cases}{{a}=?}\\{{b}=?}\end{cases} \\ $$

Question Number 161089 Answers: 3 Comments: 0

prove that I= ∫_0 ^( (π/2)) ln ( 1+ sin (2 α )) dα = 2G − π ln ((√2) ) G: catalan constant

$$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:−\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$

Question Number 161084 Answers: 0 Comments: 0

Question Number 161079 Answers: 1 Comments: 3

Question Number 161076 Answers: 1 Comments: 0

Ω = ∫_0 ^( ∞) ((ln (1+ x ))/((1+ x^( 2) )^( 2) )) dx = ? −−−−−−−−−−−−

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\:\left(\mathrm{1}+\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:−−−−−−−−−−−− \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 161075 Answers: 0 Comments: 0

simplify Σ_(n=1) ^∞ (( n)/(( n^( 2) −(( 1)/4) )^( 3) )) = ?

$$ \\ $$$$\:\:\:\:{simplify} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:{n}}{\left(\:{n}^{\:\mathrm{2}} −\frac{\:\mathrm{1}}{\mathrm{4}}\:\right)^{\:\mathrm{3}} }\:=\:? \\ $$$$ \\ $$

Question Number 161071 Answers: 2 Comments: 0

For a,b,c > 0 . Find (x,y,z) that satisfy this equation system ax + by = (x−y)^2 by + cz = (y−z)^2 cz + ax = (z−x)^2

$${For}\:\:{a},{b},{c}\:>\:\mathrm{0}\:. \\ $$$${Find}\:\:\left({x},{y},{z}\right)\:\:{that}\:\:{satisfy}\:\:{this}\:\:{equation}\:\:{system}\: \\ $$$$\:\:\:{ax}\:+\:{by}\:=\:\left({x}−{y}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{by}\:+\:{cz}\:=\:\left({y}−{z}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:{cz}\:+\:{ax}\:=\:\left({z}−{x}\right)^{\mathrm{2}} \\ $$$$ \\ $$

Question Number 161068 Answers: 2 Comments: 0

∫ ((2x)/((1−x^2 )(√(x^4 −1)))) dx =?

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

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