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Question Number 162298 Answers: 1 Comments: 0

find ∫_0 ^1 lnx ln(1−x^3 )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{lnx}\:\mathrm{ln}\left(\mathrm{1}−\mathrm{x}^{\mathrm{3}} \right)\mathrm{dx} \\ $$

Question Number 162297 Answers: 1 Comments: 0

find ∫_0 ^1 ln(1−x)ln(1+x)dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 162253 Answers: 1 Comments: 0

The tangent of a parabola y^2 =4ax at the point P (ap^2 , 2ap) intersects the line x+a=0 at T . (i) If M is the midpoint of PT , find the coordinates of M in terms of a and p. (ii) Prove that the equation of locus of M is y^2 (2x+a)=a(3x+a)^2

$$\mathrm{The}\:\mathrm{tangent}\:\mathrm{of}\:\mathrm{a}\:\mathrm{parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{ax}\:\mathrm{at}\:\mathrm{the}\:\mathrm{point} \\ $$$${P}\:\left({ap}^{\mathrm{2}} ,\:\mathrm{2}{ap}\right)\:\mathrm{intersects}\:\mathrm{the}\:\mathrm{line}\:{x}+{a}=\mathrm{0}\:\mathrm{at}\:{T}\:. \\ $$$$\left(\mathrm{i}\right)\:\mathrm{If}\:{M}\:\mathrm{is}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of}\:{PT}\:,\:\mathrm{find}\:\mathrm{the}\: \\ $$$$\:\:\:\:\:\:\mathrm{coordinates}\:\mathrm{of}\:{M}\:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{a}\:\mathrm{and}\:{p}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{locus}\:\mathrm{of}\:{M}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:{y}^{\mathrm{2}} \left(\mathrm{2}{x}+{a}\right)={a}\left(\mathrm{3}{x}+{a}\right)^{\mathrm{2}} \\ $$

Question Number 162249 Answers: 1 Comments: 0

Question Number 162243 Answers: 2 Comments: 0

∫_( 0) ^( 1) ((((ln x)^4 )/( (√x) ))) dx =?

$$\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\left(\frac{\left(\mathrm{ln}\:{x}\right)^{\mathrm{4}} }{\:\sqrt{{x}}\:}\right)\:{dx}\:=? \\ $$

Question Number 162240 Answers: 1 Comments: 0

∫((2x−5)/(x^2 +4x+5))dx

$$\int\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{5}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{4}\boldsymbol{{x}}+\mathrm{5}}\boldsymbol{{dx}} \\ $$

Question Number 162238 Answers: 1 Comments: 0

∫_0 ^1 (1/(x^7 +1))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\boldsymbol{\mathrm{x}}^{\mathrm{7}} +\mathrm{1}}\boldsymbol{\mathrm{dx}}=? \\ $$

Question Number 162234 Answers: 0 Comments: 0

Σ_(p=0) ^n (−1)^p C_p ^(1/2) (−1)^(n−p) C_(n−p) ^(1/2) = ???? Please Help me(Aidez moi)

$$ \\ $$$$\:\:\:\:\:\:\underset{{p}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{p}} {C}_{{p}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \left(−\mathrm{1}\right)^{{n}−{p}} {C}_{{n}−{p}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\:\:\:???? \\ $$$$\:\:\:\:\:\:\:{Please}\:{Help}\:{me}\left({Aidez}\:{moi}\right) \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 162229 Answers: 1 Comments: 0

Question Number 162287 Answers: 0 Comments: 4

lim_(n→∞) ∫_0 ^1 ((x^n +(1−x)^n ))^(1/n) dx=?

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt[{\mathrm{n}}]{\mathrm{x}^{\mathrm{n}} +\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{n}} }\mathrm{dx}=? \\ $$

Question Number 162219 Answers: 2 Comments: 0

Ω=∫_0 ^1 ((log(1+x^7 ))/(1+x^7 ))dx=?

$$\Omega=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{log}\left(\mathrm{1}+{x}^{\mathrm{7}} \right)}{\mathrm{1}+{x}^{\mathrm{7}} }{dx}=? \\ $$

Question Number 162209 Answers: 1 Comments: 2

Question Number 162190 Answers: 1 Comments: 0

⌊ ((3^2 +1)/(3^2 −1)) + ((3^3 +1)/(3^3 −1)) + ((3^4 +1)/(3^4 −1)) + …+ ((3^(2017) +1)/(3^(2017) −1)) ⌋ = ?

$$\lfloor\:\frac{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{3}} +\mathrm{1}}{\mathrm{3}^{\mathrm{3}} −\mathrm{1}}\:+\:\frac{\mathrm{3}^{\mathrm{4}} +\mathrm{1}}{\mathrm{3}^{\mathrm{4}} −\mathrm{1}}\:+\:\ldots+\:\frac{\mathrm{3}^{\mathrm{2017}} +\mathrm{1}}{\mathrm{3}^{\mathrm{2017}} −\mathrm{1}}\:\rfloor\:=\:\:? \\ $$

Question Number 162189 Answers: 1 Comments: 2

show the converge^ nce and calculate ∫_(−1) ^1 (√((1−t)/(1+t))) dt

$${show}\:{the}\:{converge}^{} {nce}\:{and}\:{calculate} \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}−{t}}{\mathrm{1}+{t}}}\:{dt} \\ $$

Question Number 162187 Answers: 0 Comments: 1

x^y =9 ((32y))^(1/y) =2x^2 (x,y)=? sulotion =?

$${x}^{{y}} =\mathrm{9} \\ $$$$\sqrt[{{y}}]{\mathrm{32}{y}}=\mathrm{2}{x}^{\mathrm{2}} \:\:\:\:\:\:\:\:\:\left({x},{y}\right)=? \\ $$$${sulotion}\:=? \\ $$$$ \\ $$

Question Number 162183 Answers: 1 Comments: 0

Question Number 162182 Answers: 1 Comments: 0

lim_(x→∞) (((x−(√(x^2 −3x+2)) )^(2022) +(x−(√(x^2 −5)) )^(2022) )/x^(2022) ) = ?

$$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\left(\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}}\:\right)^{\mathrm{2022}} +\left(\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{5}}\:\right)^{\mathrm{2022}} }{\mathrm{x}^{\mathrm{2022}} }\:=\:? \\ $$

Question Number 162181 Answers: 1 Comments: 1

Solve the system of equations {: ((x^4 −2x+y=y^4 )),(((x^2 −y^2 )^3 = 3)) }

$$\:\:\:\mathrm{Solve}\:\mathrm{the}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}\: \\ $$$$\:\:\:\:\:\left.\begin{matrix}{\mathrm{x}^{\mathrm{4}} −\mathrm{2x}+\mathrm{y}=\mathrm{y}^{\mathrm{4}} }\\{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)^{\mathrm{3}} \:=\:\mathrm{3}}\end{matrix}\right\}\: \\ $$

Question Number 162177 Answers: 1 Comments: 0

𝛗 = ∫_0 ^( 1) (( ln^( 2) ( x ). Li_( 2) (x ))/x^ ) dx =?

$$ \\ $$$$\:\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{ln}^{\:\mathrm{2}} \left(\:{x}\:\right).\:\mathrm{Li}_{\:\mathrm{2}} \:\left({x}\:\right)}{{x}^{\:} }\:{dx}\:=? \\ $$

Question Number 162174 Answers: 0 Comments: 1

x^( 2) − 4x −1=0 α , β are roots α^( 3) + 17β +5 =? −−−solution−−− α is root ⇒ α^( 2) −4α −1=0 ⇒ α^( 2) = 4α +1 ✓ α^( 3) + 17β +5 = α . α^( 2) +17β +5 = α ( 4α +1 )+ 17β +5 = 4α^( 2) + α + 17β +5 = 4 (4α +1 )+α +17β +5=17(α+β)+9 = 17S +9= 17 (4 )+9=77

$$ \\ $$$$\:\:\:\:{x}^{\:\mathrm{2}} −\:\mathrm{4}{x}\:−\mathrm{1}=\mathrm{0}\:\: \\ $$$$\:\:\:\:\:\alpha\:,\:\beta\:\:{are}\:{roots}\: \\ $$$$\:\:\:\:\:\alpha^{\:\mathrm{3}} \:+\:\mathrm{17}\beta\:+\mathrm{5}\:=? \\ $$$$\:\:−−−{solution}−−− \\ $$$$\:\:\:\alpha\:\:\:{is}\:{root}\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}} −\mathrm{4}\alpha\:−\mathrm{1}=\mathrm{0} \\ $$$$\:\:\:\:\:\Rightarrow\:\alpha^{\:\mathrm{2}} =\:\mathrm{4}\alpha\:+\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\alpha^{\:\mathrm{3}} +\:\mathrm{17}\beta\:+\mathrm{5}\:=\:\alpha\:.\:\alpha^{\:\mathrm{2}} +\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\alpha\:\left(\:\mathrm{4}\alpha\:+\mathrm{1}\:\right)+\:\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\mathrm{4}\alpha^{\:\mathrm{2}} +\:\alpha\:+\:\mathrm{17}\beta\:+\mathrm{5} \\ $$$$\:\:=\:\mathrm{4}\:\left(\mathrm{4}\alpha\:+\mathrm{1}\:\right)+\alpha\:+\mathrm{17}\beta\:+\mathrm{5}=\mathrm{17}\left(\alpha+\beta\right)+\mathrm{9} \\ $$$$\:\:=\:\mathrm{17S}\:+\mathrm{9}=\:\mathrm{17}\:\left(\mathrm{4}\:\right)+\mathrm{9}=\mathrm{77} \\ $$$$ \\ $$

Question Number 162171 Answers: 0 Comments: 0

Question Number 162169 Answers: 2 Comments: 0

determinant ((( determinant ((( x+y+x^2 y^2 =586_(x=?,y=? ) ^(x,y∈Z ) )))_ ^ _() ^(•) )))

$$ \\ $$$$\:\:\:\:\:\:\:\begin{array}{|c|}{\overset{\bullet} {\:\:\:\:\:\begin{array}{|c|}{\:\:\:\underset{{x}=?,{y}=?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {\overset{{x},{y}\in\mathbb{Z}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:} {{x}+{y}+{x}^{\mathrm{2}} {y}^{\mathrm{2}} =\mathrm{586}}}\:\:}\\\hline\end{array}_{} ^{} }\:\:\:\:}\\\hline\end{array} \\ $$$$ \\ $$

Question Number 162168 Answers: 0 Comments: 0

Solve the integro−differential equation: i(t) + 4(di/dt) + ∫i(t)dt = 2 cos (3t+ 60°) where i(t) is a sinulsodial current.

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{integro}−\mathrm{differential} \\ $$$$\mathrm{equation}: \\ $$$$\:{i}\left({t}\right)\:+\:\mathrm{4}\frac{{di}}{{dt}}\:+\:\int{i}\left({t}\right){dt}\:=\:\mathrm{2}\:\mathrm{cos}\:\left(\mathrm{3}{t}+\:\mathrm{60}°\right) \\ $$$$\mathrm{where}\:{i}\left({t}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{sinulsodial}\:\mathrm{current}. \\ $$

Question Number 162151 Answers: 1 Comments: 0

Question Number 162116 Answers: 2 Comments: 0

A function f is such that f : R → R where f(xy+1) = f(x)∙f(y)−f(y)−x+2 , ∀x,y ∈ R . Find value of 10∙f(2017)+f(0) .

$${A}\:{function}\:\:{f}\:\:\:{is}\:\:{such}\:\:{that}\:\:{f}\::\:\mathbb{R}\:\rightarrow\:\mathbb{R}\:\:{where} \\ $$$$\:\:\:{f}\left({xy}+\mathrm{1}\right)\:=\:{f}\left({x}\right)\centerdot{f}\left({y}\right)−{f}\left({y}\right)−{x}+\mathrm{2}\:\:,\:\:\forall{x},{y}\:\in\:\mathbb{R}\:. \\ $$$${Find}\:\:{value}\:\:{of}\:\:\mathrm{10}\centerdot{f}\left(\mathrm{2017}\right)+{f}\left(\mathrm{0}\right)\:. \\ $$

Question Number 162112 Answers: 1 Comments: 0

∫(( cos(x))/((1−cos(x))^2 ))dx=?

$$\int\frac{\:\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)}{\left(\mathrm{1}−\boldsymbol{{cos}}\left(\boldsymbol{{x}}\right)\right)^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$

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