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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.

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

Question Number 142307    Answers: 1   Comments: 0

lim_(x→0) ((tan (tan x)−tan (tan (tan x)))/(tan x∙tan (tan x)∙tan (tan (tan x))))=?

$$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{tan}\:\left(\mathrm{tan}\:\mathrm{x}\right)−\mathrm{tan}\:\left(\mathrm{tan}\:\left(\mathrm{tan}\:\mathrm{x}\right)\right)}{\mathrm{tan}\:\mathrm{x}\centerdot\mathrm{tan}\:\left(\mathrm{tan}\:\mathrm{x}\right)\centerdot\mathrm{tan}\:\left(\mathrm{tan}\:\left(\mathrm{tan}\:\mathrm{x}\right)\right)}=? \\ $$

Question Number 142306    Answers: 0   Comments: 0

Question Number 142277    Answers: 2   Comments: 0

lim_(x→0) ((1−e^(sin x ln (cos x)) )/x^3 ) =?

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

Question Number 142276    Answers: 1   Comments: 0

∫_0 ^( 2) (√(1+e^(2x) ))dx

$$\int_{\mathrm{0}} ^{\:\mathrm{2}} \sqrt{\mathrm{1}+{e}^{\mathrm{2}{x}} }{dx} \\ $$

Question Number 142275    Answers: 0   Comments: 0

.......Advanced ...∗∗∗∗∗ ...Integral...... Prove that :: Φ :=∫_0 ^( 1) ((1−x)/((1−x+x^2 )log(x)))dx= proof:: Φ:=∫_0 ^( 1) ((1−x^2 )/((1−x^3 )log(x)))dx f (a):= ∫_0 ^( 1) ((1−x^a )/((1−x^3 )log(x))) Φ := f (2) ........✓ f ′(a):=∫_0 ^( 1) ((−x^a log(x))/((1−x^3 )log(x)))=∫_0 ^( 1) ((−x^a )/(1−x^3 ))dx (★) (★):: x^3 =y ⇒ f ′(a):=(1/3)∫_0 ^( 1) ((y^((−2)/3) −y^((a/3)−(2/3)) )/(1−y))dy :=(1/3)∫_0 ^( 1) ((y^((−2)/3) −1+1−y^((a/3)−(1/3)) )/(1−y))dy :=(1/3)(ψ((a/3)+(2/3))−ψ((2/3))) f (a):=log(Γ((a/3)+(2/3)))−(a/3)ψ((2/3))+C f (0):=0=log(Γ((2/3)))+C C :=−log(Γ((2/3))) Φ:= f (2)=log(Γ((4/3)))−(2/3) ψ((2/3))−log(Γ((2/3))) :=log(((Γ((4/3)))/(Γ((2/3)))))−(2/3)ψ((2/3)) ....✓

$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:.......{Advanced}\:\:...\ast\ast\ast\ast\ast\:...{Integral}...... \\ $$$$\:\:\:\:\:\:{Prove}\:\:{that}\:::\:\:\:\Phi\::=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}}{\left(\mathrm{1}−{x}+{x}^{\mathrm{2}} \right){log}\left({x}\right)}{dx}= \\ $$$$\:\:\:{proof}:: \\ $$$$\:\:\:\:\:\:\Phi:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}^{\mathrm{2}} }{\left(\mathrm{1}−{x}^{\mathrm{3}} \right){log}\left({x}\right)}{dx} \\ $$$$\:\:\:\:\:\:{f}\:\left({a}\right):=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}−{x}^{{a}} }{\left(\mathrm{1}−{x}^{\mathrm{3}} \right){log}\left({x}\right)} \\ $$$$\:\:\:\:\:\:\:\Phi\::=\:{f}\:\left(\mathrm{2}\right)\:........\checkmark \\ $$$$\:\:\:\:\:\:\:{f}\:'\left({a}\right):=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{−{x}^{{a}} {log}\left({x}\right)}{\left(\mathrm{1}−{x}^{\mathrm{3}} \right){log}\left({x}\right)}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{−{x}^{{a}} }{\mathrm{1}−{x}^{\mathrm{3}} }{dx}\:\:\left(\bigstar\right) \\ $$$$\:\:\:\:\left(\bigstar\right)::\:\:{x}^{\mathrm{3}} ={y}\:\Rightarrow\:{f}\:'\left({a}\right):=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{y}^{\frac{−\mathrm{2}}{\mathrm{3}}} −{y}^{\frac{{a}}{\mathrm{3}}−\frac{\mathrm{2}}{\mathrm{3}}} }{\mathrm{1}−{y}}{dy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\::=\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{y}^{\frac{−\mathrm{2}}{\mathrm{3}}} −\mathrm{1}+\mathrm{1}−{y}^{\frac{{a}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{3}}} }{\mathrm{1}−{y}}{dy} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\::=\frac{\mathrm{1}}{\mathrm{3}}\left(\psi\left(\frac{{a}}{\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{3}}\right)−\psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right) \\ $$$$\:\:\:\:\:\:\:{f}\:\left({a}\right):={log}\left(\Gamma\left(\frac{{a}}{\mathrm{3}}+\frac{\mathrm{2}}{\mathrm{3}}\right)\right)−\frac{{a}}{\mathrm{3}}\psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)+{C} \\ $$$$\:\:\:\:\:\:\:\:{f}\:\left(\mathrm{0}\right):=\mathrm{0}={log}\left(\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right)+{C} \\ $$$$\:\:\:\:\:\:\:\:\:{C}\::=−{log}\left(\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\Phi:=\:{f}\:\left(\mathrm{2}\right)={log}\left(\Gamma\left(\frac{\mathrm{4}}{\mathrm{3}}\right)\right)−\frac{\mathrm{2}}{\mathrm{3}}\:\psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)−{log}\left(\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\::={log}\left(\frac{\Gamma\left(\frac{\mathrm{4}}{\mathrm{3}}\right)}{\Gamma\left(\frac{\mathrm{2}}{\mathrm{3}}\right)}\right)−\frac{\mathrm{2}}{\mathrm{3}}\psi\left(\frac{\mathrm{2}}{\mathrm{3}}\right)\:....\checkmark \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 142273    Answers: 0   Comments: 1

∫_(1/3) ^3 ((x+sin (x^2 −(1/x^2 )))/(x(2+cos (x+(1/x))))) dx ?

$$\:\:\underset{\frac{\mathrm{1}}{\mathrm{3}}} {\overset{\mathrm{3}} {\int}}\:\frac{{x}+\mathrm{sin}\:\left({x}^{\mathrm{2}} −\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}{{x}\left(\mathrm{2}+\mathrm{cos}\:\left({x}+\frac{\mathrm{1}}{{x}}\right)\right)}\:{dx}\:? \\ $$

Question Number 142164    Answers: 0   Comments: 1

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