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Question Number 161818 Answers: 1 Comments: 0

Show that: Φ =∫_( 0) ^( 1) (√((1 - x^2 )/(1 + x^2 ))) dx = ((√π)/4) (((Γ((1/4)))/(Γ((3/4)))) - 4 ((Γ((3/4)))/(Γ((1/4))))) where: Γ-Gamma function

$$\mathrm{Show}\:\mathrm{that}: \\ $$$$\Phi\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\sqrt{\frac{\mathrm{1}\:-\:\mathrm{x}^{\mathrm{2}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} }}\:\mathrm{dx}\:=\:\frac{\sqrt{\pi}}{\mathrm{4}}\:\left(\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}\:-\:\mathrm{4}\:\frac{\Gamma\left(\frac{\mathrm{3}}{\mathrm{4}}\right)}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)}\right) \\ $$$$\mathrm{where}:\:\Gamma-\mathrm{Gamma}\:\mathrm{function} \\ $$

Question Number 161817 Answers: 1 Comments: 0

Question Number 161815 Answers: 1 Comments: 0

Question Number 161811 Answers: 3 Comments: 0

Question Number 161800 Answers: 1 Comments: 2

((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!))=108 n=?

$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!}=\mathrm{108} \\ $$$${n}=? \\ $$

Question Number 161801 Answers: 0 Comments: 0

lim_(n→∞) ∫_n ^(n(cos (1/n))^n ) (1−(4/x))^x dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{n}} ^{\mathrm{n}\left(\mathrm{cos}\:\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} } \left(\mathrm{1}−\frac{\mathrm{4}}{\mathrm{x}}\right)^{\mathrm{x}} \mathrm{dx}=? \\ $$

Question Number 161802 Answers: 1 Comments: 0

Question Number 161786 Answers: 2 Comments: 0

logx_(4x) +logx_(x/2) =2 solve for x=?

$${log}\underset{\mathrm{4}{x}} {{x}}+{log}\underset{\frac{{x}}{\mathrm{2}}} {{x}}=\mathrm{2} \\ $$$${solve}\:\:\:{for}\:\:\:{x}=? \\ $$

Question Number 161783 Answers: 1 Comments: 5

Question Number 161773 Answers: 3 Comments: 1

Question Number 161770 Answers: 2 Comments: 0

lim_(x→0) ((((√(1+2x^2 ))+2x)^(2021) −((√(1+2x^2 ))−2x)^(2021) )/x)

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }+\mathrm{2}{x}\right)^{\mathrm{2021}} −\left(\sqrt{\mathrm{1}+\mathrm{2}{x}^{\mathrm{2}} }−\mathrm{2}{x}\right)^{\mathrm{2021}} }{{x}} \\ $$

Question Number 161760 Answers: 1 Comments: 0

lim_(x→0) ((tan x)/( (((1−cos x)^2 ))^(1/3) )) =?

$$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:{x}}{\:\sqrt[{\mathrm{3}}]{\left(\mathrm{1}−\mathrm{cos}\:{x}\right)^{\mathrm{2}} }}\:=? \\ $$

Question Number 161755 Answers: 2 Comments: 0

prove that Σ_(n=0) ^∞ (((−1)^n )/(n+1)) = ln2

$${prove}\:{that}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:=\:{ln}\mathrm{2} \\ $$

Question Number 161750 Answers: 0 Comments: 2

differenciate xsin xcos x

$${differenciate}\:{x}\mathrm{sin}\:{x}\mathrm{cos}\:{x} \\ $$

Question Number 161748 Answers: 0 Comments: 4

⌋

$$\rfloor \\ $$

Question Number 161747 Answers: 2 Comments: 0

lim_(x→0) ((cos^3 (2x)−cos (x))/(cos^2 (4x)−cos (2x))) =?

$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:^{\mathrm{3}} \left(\mathrm{2}{x}\right)−\mathrm{cos}\:\left({x}\right)}{\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{4}{x}\right)−\mathrm{cos}\:\left(\mathrm{2}{x}\right)}\:=? \\ $$

Question Number 161745 Answers: 1 Comments: 1

Prove that: 3 (√e) = (1/3) Σ_(k=1) ^∞ [ Σ_(n=0) ^∞ (1/(n!)) ]k2^(-k)

$$\mathrm{Prove}\:\mathrm{that}: \\ $$$$\mathrm{3}\:\sqrt{\mathrm{e}}\:=\:\frac{\mathrm{1}}{\mathrm{3}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left[\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}!}\:\right]\mathrm{k2}^{-\boldsymbol{\mathrm{k}}} \\ $$

Question Number 161744 Answers: 1 Comments: 1

Solve for real numbers: (x/y) + (5/x) + ((y - 5)/5) = ((y + x)/(y + 5)) + ((5 + y)/(5 + x))

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\:+\:\frac{\mathrm{y}\:-\:\mathrm{5}}{\mathrm{5}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{x}}{\mathrm{y}\:+\:\mathrm{5}}\:+\:\frac{\mathrm{5}\:+\:\mathrm{y}}{\mathrm{5}\:+\:\mathrm{x}} \\ $$

Question Number 161742 Answers: 2 Comments: 1

Question Number 161733 Answers: 1 Comments: 0

Question Number 161723 Answers: 4 Comments: 0

Calculate lim_(x→0) (((1+x.2^x )/(1+x.3^x )))^(1/x^2 ) lim_(x→0) [2e^(x/(x+1)) −1]^((x^2 +1)/x) lim_(x→a) ((x^x −a^a )/(x−a))

$${Calculate} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{1}+{x}.\mathrm{2}^{{x}} }{\mathrm{1}+{x}.\mathrm{3}^{{x}} }\right)^{\frac{\mathrm{1}}{{x}^{\mathrm{2}} }} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\mathrm{2}{e}^{\frac{{x}}{{x}+\mathrm{1}}} −\mathrm{1}\right]^{\frac{{x}^{\mathrm{2}} +\mathrm{1}}{{x}}} \\ $$$$\underset{{x}\rightarrow{a}} {\mathrm{lim}}\frac{{x}^{{x}} −{a}^{{a}} }{{x}−{a}} \\ $$

Question Number 161706 Answers: 1 Comments: 0

J =∫_0 ^( 1) (( 1−x)/(( 1+x +x^( 2) + x^( 3) )ln(x))) dx=?

$$\: \\ $$$$\mathrm{J}\:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{1}−{x}}{\left(\:\mathrm{1}+{x}\:+{x}^{\:\mathrm{2}} +\:{x}^{\:\mathrm{3}} \:\right){ln}\left({x}\right)}\:{dx}=? \\ $$$$ \\ $$

Question Number 161704 Answers: 2 Comments: 1

let n∈N fixed , solve for real numbers [x]{x}=nx

$$\mathrm{let}\:\:\mathrm{n}\in\mathbb{N}\:\:\mathrm{fixed}\:,\:\mathrm{solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\left[\mathrm{x}\right]\left\{\mathrm{x}\right\}=\mathrm{nx} \\ $$

Question Number 161698 Answers: 2 Comments: 6

sin (x+y)=sin x+sin y

$$\:\:\mathrm{sin}\:\left({x}+{y}\right)=\mathrm{sin}\:{x}+\mathrm{sin}\:{y} \\ $$

Question Number 161697 Answers: 1 Comments: 0

Question Number 161687 Answers: 1 Comments: 0

nature of the integral ∫_0 ^1 (1/(t^2 (√(1−t))))dt

$${nature}\:{of}\:{the}\:{integral} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{{t}^{\mathrm{2}} \sqrt{\mathrm{1}−{t}}}{dt} \\ $$

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