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Question Number 142358 Answers: 1 Comments: 0

∫_0 ^∞ x^(n−1) log_e (1−x)dx

$$\int_{\mathrm{0}} ^{\infty} {x}^{{n}−\mathrm{1}} {log}_{{e}} \left(\mathrm{1}−{x}\right){dx} \\ $$

Question Number 142356 Answers: 0 Comments: 0

Question Number 142351 Answers: 1 Comments: 0

lim_(x→0) ((((27+x))^(1/(3 )) −((27−x))^(1/(3 )) )/( (x^2 )^(1/(3 )) + (x^3 )^(1/(4 )) )) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{3}\:}]{\mathrm{27}+{x}}−\sqrt[{\mathrm{3}\:}]{\mathrm{27}−{x}}}{\:\sqrt[{\mathrm{3}\:}]{{x}^{\mathrm{2}} }\:+\:\sqrt[{\mathrm{4}\:}]{{x}^{\mathrm{3}} }}\:=? \\ $$

Question Number 142347 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((x^2 logx)/(x^6 +1))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{x}^{\mathrm{2}} \mathrm{logx}}{\mathrm{x}^{\mathrm{6}} \:+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 142346 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((log(1+x^3 ))/(1+x^4 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{log}\left(\mathrm{1}+\mathrm{x}^{\mathrm{3}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 142344 Answers: 0 Comments: 3

𝛗:=∫^( e) _(1/e) {(1/(ln(x)))+ln(ln(x))}dx

$$ \\ $$$$\:\:\:\:\:\:\boldsymbol{\phi}:=\underset{\frac{\mathrm{1}}{{e}}} {\int}^{\:{e}} \left\{\frac{\mathrm{1}}{{ln}\left({x}\right)}+{ln}\left({ln}\left({x}\right)\right)\right\}{dx} \\ $$

Question Number 142341 Answers: 1 Comments: 1

Question Number 142338 Answers: 2 Comments: 0

∫_0 ^∞ ((sinx)/x^μ )dx =?

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}^{\mu} }{dx}\:\:=?\:\:\: \\ $$

Question Number 142325 Answers: 0 Comments: 0

Given that a ≥ 1 ≥ b > 0. Prove the followings: (1) (1/2)(a−b)^2 ≤ (a−1)^2 +(1−b)^2 ≤ (a−b)^2 (2) (1/4)(a−b)^3 ≤ (a−1)^3 +(1−b)^3 ≤ (a−b)^3

$$\mathrm{Given}\:\mathrm{that}\:{a}\:\geqslant\:\mathrm{1}\:\geqslant\:{b}\:>\:\mathrm{0}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{followings}:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\left({a}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{1}−{b}\right)^{\mathrm{2}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{2}} \:\:\:\: \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{4}}\left({a}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−\mathrm{1}\right)^{\mathrm{3}} +\left(\mathrm{1}−{b}\right)^{\mathrm{3}} \:\leqslant\:\left({a}−{b}\right)^{\mathrm{3}} \\ $$$$ \\ $$

Question Number 142349 Answers: 2 Comments: 0

Σ_(n=0) ^∞ ((ζ(2n+2)(−1)^n )/4^n )=?

$$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\zeta\left(\mathrm{2}{n}+\mathrm{2}\right)\left(−\mathrm{1}\right)^{{n}} }{\mathrm{4}^{{n}} }=? \\ $$

Question Number 142348 Answers: 1 Comments: 0

(((4−(√(15))))^(1/6) /( (√(4−(√(15)))) ∙ ((4+(√(15))))^(1/3) )) = ?

$$\frac{\sqrt[{\mathrm{6}}]{\mathrm{4}−\sqrt{\mathrm{15}}}}{\:\sqrt{\mathrm{4}−\sqrt{\mathrm{15}}}\:\centerdot\:\sqrt[{\mathrm{3}}]{\mathrm{4}+\sqrt{\mathrm{15}}}}\:=\:? \\ $$

Question Number 142318 Answers: 1 Comments: 0

Nice...≽≽≽∗∗∗≼≼≼...Calculus Ω:=∫_0 ^( 1) (((1−(x)^(1/3) )(1−((x ))^(1/5) )(1−(x)^(1/7) ))/(ln( ((x ))^(1/3) ))) dx=? ....m.n

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Nice}...\succcurlyeq\succcurlyeq\succcurlyeq\ast\ast\ast\preccurlyeq\preccurlyeq\preccurlyeq...{Calculus} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Omega:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left(\mathrm{1}−\sqrt[{\mathrm{3}}]{{x}}\:\right)\left(\mathrm{1}−\sqrt[{\mathrm{5}}]{{x}\:}\:\right)\left(\mathrm{1}−\sqrt[{\mathrm{7}}]{{x}}\:\right)}{{ln}\left(\:\sqrt[{\mathrm{3}}]{{x}\:\:}\:\right)}\:{dx}=? \\ $$$$\:\:\:\:\:\:\:....{m}.{n} \\ $$

Question Number 142333 Answers: 2 Comments: 1

Question Number 142329 Answers: 2 Comments: 0

lim_(x→∞) (((x!)/x^x ))^(1/x)

$$\:{lim}_{{x}\rightarrow\infty} \:\left(\frac{{x}!}{{x}^{{x}} }\right)^{\frac{\mathrm{1}}{{x}}} \\ $$

Question Number 142326 Answers: 2 Comments: 0

Question Number 142310 Answers: 2 Comments: 0

Show that for n∈ N, A_n =n^2 (n^2 −1) is divisible by 12

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{for}\:\mathrm{n}\in\:\mathbb{N},\:\mathrm{A}_{\mathrm{n}} =\mathrm{n}^{\mathrm{2}} \left(\mathrm{n}^{\mathrm{2}} −\mathrm{1}\right) \\ $$$$\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{12} \\ $$

Question Number 142309 Answers: 1 Comments: 0

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

$$\underset{\mathrm{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\mathrm{x}^{\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{x}} } −\left(\mathrm{sin}\:\mathrm{x}\right)^{\mathrm{x}^{\mathrm{sin}\:\mathrm{x}} } }{\mathrm{x}^{\mathrm{3}} }=? \\ $$

Question Number 142308 Answers: 0 Comments: 0

lim_(x→0) ((lnlnln[x+(1+x)^(((1+x)^(1/x) )/x) ]+x(1−(1/e^(e+1) )))/x^2 )=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{lnlnln}\left[\mathrm{x}+\left(\mathrm{1}+\mathrm{x}\right)^{\frac{\left(\mathrm{1}+\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} }{\mathrm{x}}} \right]+\mathrm{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{e}^{\mathrm{e}+\mathrm{1}} }\right)}{\mathrm{x}^{\mathrm{2}} }=? \\ $$

Question Number 142304 Answers: 0 Comments: 0

Question Number 142305 Answers: 1 Comments: 0

Question Number 142301 Answers: 1 Comments: 0

Question Number 142300 Answers: 0 Comments: 0

lim_(x→0) (((e^(sin x) +sin x)^(1/(sin x)) −(e^(tan x) +tan x)^(1/(tan x)) )/x^3 )=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{e}^{\mathrm{sin}\:\mathrm{x}} +\mathrm{sin}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{sin}\:\mathrm{x}}} −\left(\mathrm{e}^{\mathrm{tan}\:\mathrm{x}} +\mathrm{tan}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{tan}\:\mathrm{x}}} }{\mathrm{x}^{\mathrm{3}} }=? \\ $$

Question Number 142299 Answers: 0 Comments: 0

Question Number 142290 Answers: 1 Comments: 0

evaluate: Θ:=Σ_(n=1) ^∞ ((ζ(n+1)−1)/(n+1)) =?

$$\:\:\:{evaluate}: \\ $$$$\:\:\:\:\:\:\Theta:=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\zeta\left({n}+\mathrm{1}\right)−\mathrm{1}}{{n}+\mathrm{1}}\:=? \\ $$$$\:\:\:\:\: \\ $$

Question Number 142285 Answers: 2 Comments: 0

L(((1+2bt)/( (√t)))e^(bt) )(s)=?

$$\mathscr{L}\left(\frac{\mathrm{1}+\mathrm{2bt}}{\:\sqrt{\mathrm{t}}}\mathrm{e}^{\mathrm{bt}} \right)\left(\mathrm{s}\right)=? \\ $$

Question Number 142282 Answers: 1 Comments: 0

Three interior angles of a polygon are 160° each. If the other interior angles are 120° each, find the number of sides of the polygon.

$$\mathrm{Three}\:\mathrm{interior}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{a}\:\mathrm{polygon}\:\mathrm{are}\:\mathrm{160}° \\ $$$$\mathrm{each}.\:\mathrm{If}\:\mathrm{the}\:\mathrm{other}\:\mathrm{interior}\:\mathrm{angles}\:\mathrm{are}\:\mathrm{120}°\:\mathrm{each}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{sides}\:\mathrm{of}\:\mathrm{the}\:\mathrm{polygon}. \\ $$

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