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AllQuestion and Answers: Page 587

Question Number 161186 Answers: 2 Comments: 0

Question Number 161181 Answers: 0 Comments: 1

x=cot^(−1) ((√(cos θ)))−tan^(−1) ((√(cos θ))) sin x=?

$$\:\:{x}=\mathrm{cot}^{−\mathrm{1}} \left(\sqrt{\mathrm{cos}\:\theta}\right)−\mathrm{tan}^{−\mathrm{1}} \left(\sqrt{\mathrm{cos}\:\theta}\right) \\ $$$$\:\mathrm{sin}\:{x}=? \\ $$

Question Number 161179 Answers: 0 Comments: 1

Question Number 161178 Answers: 1 Comments: 0

∫^∞ _2 ((arctg(x))/(arctg((x/2))))dx=???

$$\underset{\mathrm{2}} {\int}^{\infty} \frac{\boldsymbol{{arctg}}\left(\boldsymbol{{x}}\right)}{\boldsymbol{{arctg}}\left(\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)}\boldsymbol{{dx}}=??? \\ $$

Question Number 161177 Answers: 1 Comments: 0

Question Number 161176 Answers: 0 Comments: 0

calculate Θ := Σ_(n=1) ^∞ (( (−1 )^( n−1) )/(n ( n + (1/3) ))) =? ■ m.n −−−−−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:{calculate}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Theta\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\:\right)^{\:{n}−\mathrm{1}} }{{n}\:\left(\:{n}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\right)}\:=?\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:\:−−−−−−−−−−−−− \\ $$$$ \\ $$

Question Number 161169 Answers: 2 Comments: 0

if 9x^2 +(1/x^2 )=3 then 27x^3 +(1/x^3 )=?

$${if}\:\mathrm{9}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{{x}^{\mathrm{2}} }=\mathrm{3} \\ $$$${then} \\ $$$$\:\mathrm{27}{x}^{\mathrm{3}} +\frac{\mathrm{1}}{{x}^{\mathrm{3}} }=? \\ $$

Question Number 161168 Answers: 2 Comments: 1

Show that U_n =((4n−1)/(7n+3)) is convergent sequence.

$${Show}\:{that}\:{U}_{{n}} =\frac{\mathrm{4}{n}−\mathrm{1}}{\mathrm{7}{n}+\mathrm{3}}\:\:{is}\:{convergent}\: \\ $$$${sequence}. \\ $$

Question Number 161166 Answers: 1 Comments: 0

If ^4 log (x+2y) +^4 log (x−2y) = 1 . Minimum value of ∣x∣ − ∣y∣ is ... ?

$${If}\:\:\:^{\mathrm{4}} \mathrm{log}\:\left({x}+\mathrm{2}{y}\right)\:+\:^{\mathrm{4}} \mathrm{log}\:\left({x}−\mathrm{2}{y}\right)\:=\:\mathrm{1}\:. \\ $$$${Minimum}\:\:{value}\:\:{of}\:\:\mid{x}\mid\:−\:\mid{y}\mid\:\:\:{is}\:\:...\:? \\ $$

Question Number 161163 Answers: 1 Comments: 0

Question Number 161162 Answers: 0 Comments: 0

f :]0,+∞[→]0,+∞[ is convex function for n≥2 an integer , prove : (f(1)^(f(1)) f(2)^(f(2)) ...f(n)^(f(n)) )^(1/(f(1)+f(2)+...+f(n))) +(f(1)f(2)...f(n))^(1/n) ≤f(1)+f(n)

$$\left.{f}\::\right]\mathrm{0},+\infty\left[\rightarrow\right]\mathrm{0},+\infty\left[\:{is}\:{convex}\:{function}\right. \\ $$$${for}\:{n}\geqslant\mathrm{2}\:{an}\:{integer}\:,\:{prove}\:: \\ $$$$\left({f}\left(\mathrm{1}\right)^{{f}\left(\mathrm{1}\right)} {f}\left(\mathrm{2}\right)^{{f}\left(\mathrm{2}\right)} ...{f}\left({n}\right)^{{f}\left({n}\right)} \right)^{\frac{\mathrm{1}}{{f}\left(\mathrm{1}\right)+{f}\left(\mathrm{2}\right)+...+{f}\left({n}\right)}} +\left({f}\left(\mathrm{1}\right){f}\left(\mathrm{2}\right)...{f}\left({n}\right)\right)^{\frac{\mathrm{1}}{{n}}} \leqslant{f}\left(\mathrm{1}\right)+{f}\left({n}\right) \\ $$

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

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

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