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Question Number 161854 Answers: 1 Comments: 1

Question Number 161848 Answers: 0 Comments: 2

If tan (α )= 2 find the value of K=(( 1+sin( 8 α)−cos (8α ))/(1+sin( 8α ) + cos (8 α ))) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}{f}\:\:\:\:{tan}\:\left(\alpha\:\right)=\:\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:{find}\:{the}\:{value}\:{of}\: \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\mathrm{K}=\frac{\:\:\mathrm{1}+{sin}\left(\:\mathrm{8}\:\alpha\right)−{cos}\:\left(\mathrm{8}\alpha\:\right)}{\mathrm{1}+{sin}\left(\:\mathrm{8}\alpha\:\right)\:+\:{cos}\:\left(\mathrm{8}\:\alpha\:\right)}\:=? \\ $$$$ \\ $$$$ \\ $$

Question Number 161843 Answers: 0 Comments: 3

Q#161744 reposted with some change. Solve for integer numbers: (x/y) + (5/x) + ((y - 5)/5) = ((y + x)/(y + 5)) + ((5 + y)/(5 + x))

$${Q}#\mathrm{161744}\:{reposted}\:{with}\:{some}\:{change}. \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\boldsymbol{\mathrm{integer}}\:\mathrm{numbers}: \\ $$$$\frac{\mathrm{x}}{\mathrm{y}}\:+\:\frac{\mathrm{5}}{\mathrm{x}}\:+\:\frac{\mathrm{y}\:-\:\mathrm{5}}{\mathrm{5}}\:=\:\frac{\mathrm{y}\:+\:\mathrm{x}}{\mathrm{y}\:+\:\mathrm{5}}\:+\:\frac{\mathrm{5}\:+\:\mathrm{y}}{\mathrm{5}\:+\:\mathrm{x}} \\ $$

Question Number 161839 Answers: 3 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 +2x+2)^2 ))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$

Question Number 161830 Answers: 1 Comments: 0

∫(dx/( (√(1−x^(14) ))))

$$\int\frac{{dx}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{14}} }} \\ $$

Question Number 161823 Answers: 1 Comments: 0

Given that −1<x<1, find the expansion of ((3−2x)/((1+x)(4+x^2 ))) in ascending power of x, up to and including the term in x^3

$$\:{Given}\:{that}\:−\mathrm{1}<{x}<\mathrm{1},\:{find}\:{the} \\ $$$$\:{expansion}\:{of}\:\:\frac{\mathrm{3}−\mathrm{2}{x}}{\left(\mathrm{1}+{x}\right)\left(\mathrm{4}+{x}^{\mathrm{2}} \right)}\:{in} \\ $$$$\:{ascending}\:{power}\:{of}\:{x},\:{up}\:{to}\:{and} \\ $$$$\:{including}\:{the}\:{term}\:{in}\:{x}^{\mathrm{3}} \\ $$

Question Number 161822 Answers: 1 Comments: 0

Find: Ω =lim_(n→∞) Σ_(k=1) ^n [Σ_(i=1) ^k (i + (1/4))]^( -1) = ?

$$\mathrm{Find}: \\ $$$$\Omega\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\left[\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\left(\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{4}}\right)\right]^{\:-\mathrm{1}} =\:? \\ $$

Question Number 161820 Answers: 1 Comments: 0

Solve this differential equation: a ((∂L(α))/∂a) + b ((∂L(α))/∂b) = L(α) where: L(α) = ∫_( 0) ^( a) (√(a^2 sin^2 (t) + b^2 cos^2 (t))) dt

$$\mathrm{Solve}\:\mathrm{this}\:\mathrm{differential}\:\mathrm{equation}: \\ $$$${a}\:\frac{\partial{L}\left(\alpha\right)}{\partial{a}}\:+\:{b}\:\frac{\partial{L}\left(\alpha\right)}{\partial{b}}\:=\:{L}\left(\alpha\right) \\ $$$$\mathrm{where}:\:{L}\left(\alpha\right)\:=\:\underset{\:\mathrm{0}} {\overset{\:\boldsymbol{{a}}} {\int}}\sqrt{{a}^{\mathrm{2}} {sin}^{\mathrm{2}} \left({t}\right)\:+\:{b}^{\mathrm{2}} {cos}^{\mathrm{2}} \left({t}\right)}\:{dt} \\ $$

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))

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

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