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Question Number 156931    Answers: 1   Comments: 2

(1−x^2 )y′′−4xy′−(1+x^2 )y=0

$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right){y}''−\mathrm{4}{xy}'−\left(\mathrm{1}+{x}^{\mathrm{2}} \right){y}=\mathrm{0} \\ $$

Question Number 156927    Answers: 0   Comments: 0

Find: 𝛀 =lim_(n→∞) ∫_( 0) ^( 5) (((1 - x)∙x^(n+4) )/(1 + x^(3n) )) dx = ?

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{5}} {\int}}\:\frac{\left(\mathrm{1}\:-\:\mathrm{x}\right)\centerdot\mathrm{x}^{\boldsymbol{\mathrm{n}}+\mathrm{4}} }{\mathrm{1}\:+\:\mathrm{x}^{\mathrm{3}\boldsymbol{\mathrm{n}}} }\:\mathrm{dx}\:=\:? \\ $$

Question Number 156924    Answers: 1   Comments: 0

Π_(k=2) ^∞ (((k^3 −1)/(k^3 +1))) =?

$$\:\underset{{k}=\mathrm{2}} {\overset{\infty} {\prod}}\:\left(\frac{{k}^{\mathrm{3}} −\mathrm{1}}{{k}^{\mathrm{3}} +\mathrm{1}}\right)\:=? \\ $$

Question Number 156923    Answers: 1   Comments: 0

𝛀 =∫_( 0) ^( (𝛑/4)) x log (1 + tanx) dx = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {\int}}\mathrm{x}\:\mathrm{log}\:\left(\mathrm{1}\:+\:\mathrm{tan}\boldsymbol{\mathrm{x}}\right)\:\mathrm{dx}\:=\:? \\ $$

Question Number 156921    Answers: 1   Comments: 0

x^3 −4x^2 −3=0 x∈R

$${x}^{\mathrm{3}} −\mathrm{4}{x}^{\mathrm{2}} −\mathrm{3}=\mathrm{0} \\ $$$$\:{x}\in\mathbb{R} \\ $$

Question Number 156913    Answers: 1   Comments: 0

Question Number 157054    Answers: 0   Comments: 1

suppose you drop a tennis ball from a hieght of 15 feet.after the ballhits the floor it rebounds to85% of its previous height.how high will the ball rebound after its ghird bounce round tl the nearest tenth

$${suppose}\:{you}\:{drop}\:{a}\:{tennis}\:{ball}\:{from}\:{a}\:{hieght}\:{of}\:\mathrm{15}\:{feet}.{after}\:{the}\:{ballhits}\:{the}\:{floor}\:{it}\:{rebounds}\:\:{to}\mathrm{85\%}\:{of}\:{its}\:{previous}\:{height}.{how}\:{high}\:{will}\:{the}\:{ball}\:{rebound}\:{after}\:{its}\:{ghird}\:{bounce}\:{round}\:{tl}\:{the}\:{nearest}\:{tenth} \\ $$$$ \\ $$

Question Number 156911    Answers: 0   Comments: 0

Find: 𝛀 =lim_(n→∞) ∫_( 0) ^( 1) ([nx] ∙ ∣x - [x + (1/2)∣])dx [∗] - GIF

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\left(\left[\mathrm{nx}\right]\:\centerdot\:\mid\mathrm{x}\:-\:\left[\mathrm{x}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\mid\right]\right)\mathrm{dx} \\ $$$$\left[\ast\right]\:-\:\mathrm{GIF} \\ $$

Question Number 156910    Answers: 0   Comments: 2

Question Number 156909    Answers: 0   Comments: 0

Find: 𝛀 =lim_(n→∞) (n - Σ_(k=1) ^n (((e - 1)∙n)/(n + (e - 1)∙k))) = ?

$$\mathrm{Find}: \\ $$$$\boldsymbol{\Omega}\:\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{n}\:-\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\left(\mathrm{e}\:-\:\mathrm{1}\right)\centerdot\mathrm{n}}{\mathrm{n}\:+\:\left(\mathrm{e}\:-\:\mathrm{1}\right)\centerdot\mathrm{k}}\right)\:=\:? \\ $$

Question Number 156905    Answers: 2   Comments: 0

find the value of x and y, x:3:5=8:y:9

$${find}\:{the}\:{value}\:{of}\:{x}\:{and}\:{y},\:{x}:\mathrm{3}:\mathrm{5}=\mathrm{8}:{y}:\mathrm{9} \\ $$

Question Number 156904    Answers: 1   Comments: 0

find the value of x and y , x:3:5=2:y:10

$${find}\:{the}\:{value}\:{of}\:{x}\:{and}\:{y}\:,\:{x}:\mathrm{3}:\mathrm{5}=\mathrm{2}:{y}:\mathrm{10} \\ $$

Question Number 156900    Answers: 0   Comments: 0

Ω_1 = 1 - (π/2) +Σ_(n=2) ^∞ (- (1/𝛑))^n ∙ (1/(n+1)) Ω_2 = 1 - (π/2) + Σ_(n=2) ^∞ (- (1/e))^n ∙ (1/(n+1)) A) Ω_1 < Ω_2 B) Ω_1 = Ω_2 C) Ω_1 > Ω_2

$$\Omega_{\mathrm{1}} \:=\:\mathrm{1}\:-\:\frac{\pi}{\mathrm{2}}\:+\underset{\boldsymbol{\mathrm{n}}=\mathrm{2}} {\overset{\infty} {\sum}}\left(-\:\frac{\mathrm{1}}{\boldsymbol{\pi}}\right)^{\boldsymbol{\mathrm{n}}} \centerdot\:\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}} \\ $$$$\Omega_{\mathrm{2}} \:=\:\mathrm{1}\:-\:\frac{\pi}{\mathrm{2}}\:+\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{2}} {\overset{\infty} {\sum}}\left(-\:\frac{\mathrm{1}}{\boldsymbol{\mathrm{e}}}\right)^{\boldsymbol{\mathrm{n}}} \centerdot\:\frac{\mathrm{1}}{\mathrm{n}+\mathrm{1}} \\ $$$$\left.\mathrm{A}\left.\right)\left.\:\Omega_{\mathrm{1}} \:<\:\Omega_{\mathrm{2}} \:\:\:\mathrm{B}\right)\:\Omega_{\mathrm{1}} \:=\:\Omega_{\mathrm{2}} \:\:\:\mathrm{C}\right)\:\Omega_{\mathrm{1}} \:>\:\Omega_{\mathrm{2}} \\ $$

Question Number 156895    Answers: 0   Comments: 0

(dy/dx)−(x/y)+x^3 cos y = 0

$$\:\frac{{dy}}{{dx}}−\frac{{x}}{{y}}+{x}^{\mathrm{3}} \:\mathrm{cos}\:{y}\:=\:\mathrm{0} \\ $$

Question Number 156915    Answers: 1   Comments: 0

Question Number 156914    Answers: 2   Comments: 0

Question Number 156891    Answers: 1   Comments: 0

Question Number 156887    Answers: 1   Comments: 0

tan 2x tan 3x tan 5x =1

$$\:\mathrm{tan}\:\mathrm{2}{x}\:\mathrm{tan}\:\mathrm{3}{x}\:\mathrm{tan}\:\mathrm{5}{x}\:=\mathrm{1} \\ $$

Question Number 156869    Answers: 2   Comments: 0

Question Number 156867    Answers: 1   Comments: 0

Question Number 156864    Answers: 0   Comments: 4

φ := ∫_0 ^( 1) (( ln (1−x^( 2) ))/(1+ x^( 2) )) dx = proof : φ = ∫_0 ^( 1) (( ln(1−x ))/(1+x^( 2) ))dx + (π/8)ln(2) .... I= ∫_0 ^( 1) ((ln ( 1−x ))/(1+x^( 2) ))dx =^(x=tan(t)) ∫_0 ^( (π/4)) ln( cos(t)−sin(t))dt−∫_0 ^( (π/4)) ln(cos(t))dt = ∫_0 ^( (π/4)) ln((√2) )dt +∫_0 ^( (π/4)) ln(sin((π/4) −t))dt−(G/2) +(π/4)ln(2) =((3π)/8) ln(2)−(G/2) −(G/2) −(π/4) ln(2)=(π/8)ln(2)−G φ = (π/4)ln(2) − G ■ m.n

$$ \\ $$$$\:\:\:\:\phi\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\:\left(\mathrm{1}−{x}^{\:\mathrm{2}} \right)}{\mathrm{1}+\:{x}^{\:\mathrm{2}} }\:{dx}\:= \\ $$$$\:\:{proof}\:: \\ $$$$\:\:\:\:\phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\left(\mathrm{1}−{x}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx}\:+\:\frac{\pi}{\mathrm{8}}{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:\:....\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\:\left(\:\mathrm{1}−{x}\:\right)}{\mathrm{1}+{x}^{\:\mathrm{2}} }{dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{{x}={tan}\left({t}\right)} {=}\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {ln}\left(\:{cos}\left({t}\right)−{sin}\left({t}\right)\right){dt}−\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {ln}\left({cos}\left({t}\right)\right){dt} \\ $$$$\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {ln}\left(\sqrt{\mathrm{2}}\:\right){dt}\:+\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} {ln}\left({sin}\left(\frac{\pi}{\mathrm{4}}\:−{t}\right)\right){dt}−\frac{\mathrm{G}}{\mathrm{2}}\:+\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right) \\ $$$$\:\:\:=\frac{\mathrm{3}\pi}{\mathrm{8}}\:{ln}\left(\mathrm{2}\right)−\frac{\mathrm{G}}{\mathrm{2}}\:−\frac{\mathrm{G}}{\mathrm{2}}\:−\frac{\pi}{\mathrm{4}}\:{ln}\left(\mathrm{2}\right)=\frac{\pi}{\mathrm{8}}{ln}\left(\mathrm{2}\right)−\mathrm{G} \\ $$$$\:\:\:\:\phi\:=\:\frac{\pi}{\mathrm{4}}{ln}\left(\mathrm{2}\right)\:−\:\mathrm{G}\:\:\:\:\:\:\blacksquare\:{m}.{n} \\ $$

Question Number 156860    Answers: 1   Comments: 0

∫_0 ^∞ ((x)^(1/n) /(x^3 +x^2 +x+1))dx=?

$$\int_{\mathrm{0}} ^{\infty} \frac{\sqrt[{{n}}]{{x}}}{{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}}{dx}=? \\ $$

Question Number 156855    Answers: 1   Comments: 0

(dy/dx) = ((y^3 −xy^2 −x^2 y−5x^3 )/(xy^2 −x^2 y−2x^3 ))

$$\:\:\frac{{dy}}{{dx}}\:=\:\frac{{y}^{\mathrm{3}} −{xy}^{\mathrm{2}} −{x}^{\mathrm{2}} {y}−\mathrm{5}{x}^{\mathrm{3}} }{{xy}^{\mathrm{2}} −{x}^{\mathrm{2}} {y}−\mathrm{2}{x}^{\mathrm{3}} }\: \\ $$$$\: \\ $$

Question Number 156881    Answers: 0   Comments: 0

𝛀 =∫_( 0) ^( 1) ∫_( 0) ^( 1) ((log(1 - x) log(1 - y))/(1 - xy)) dxdy = ?

$$\boldsymbol{\Omega}\:=\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:\frac{\mathrm{log}\left(\mathrm{1}\:-\:\mathrm{x}\right)\:\mathrm{log}\left(\mathrm{1}\:-\:\mathrm{y}\right)}{\mathrm{1}\:-\:\mathrm{xy}}\:\mathrm{dxdy}\:=\:? \\ $$

Question Number 156880    Answers: 2   Comments: 0

𝛀 =Σ_(n=0) ^∞ Σ_(k=0) ^n (1/𝛑^n ) ∙ ((π/e))^k = ?

$$\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}=\mathrm{0}} {\overset{\infty} {\sum}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{0}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\frac{\mathrm{1}}{\boldsymbol{\pi}^{\boldsymbol{\mathrm{n}}} }\:\centerdot\:\left(\frac{\pi}{\mathrm{e}}\right)^{\boldsymbol{\mathrm{k}}} =\:? \\ $$

Question Number 156841    Answers: 1   Comments: 0

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