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Question Number 143677 Answers: 3 Comments: 0

Question Number 143671 Answers: 0 Comments: 2

Question Number 143666 Answers: 2 Comments: 1

Question Number 143653 Answers: 1 Comments: 0

Question Number 143650 Answers: 3 Comments: 0

If the function f and g are defined on the set of real numbers,are such that gof(x)=((2x−5)/(3x+7)) and g(x)=((3x+2)/(x−5)). find an expression for f(x)

$${If}\:\:{the}\:{function}\:{f}\:{and}\:{g}\:{are}\:{defined} \\ $$$${on}\:{the}\:{set}\:{of}\:{real}\:{numbers},{are}\:{such} \\ $$$${that}\:\boldsymbol{{gof}}\left(\boldsymbol{{x}}\right)=\frac{\mathrm{2}\boldsymbol{{x}}−\mathrm{5}}{\mathrm{3}\boldsymbol{{x}}+\mathrm{7}}\:\:\:{and}\: \\ $$$$\boldsymbol{{g}}\left(\boldsymbol{{x}}\right)=\frac{\mathrm{3}\boldsymbol{{x}}+\mathrm{2}}{\boldsymbol{{x}}−\mathrm{5}}. \\ $$$$\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{expression}}\:\boldsymbol{\mathrm{for}}\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}}\right) \\ $$

Question Number 143889 Answers: 2 Comments: 0

A Challanging Integral: Φ = ∫_0 ^( 1) ((log(x).log(1+x))/(1−x))dx

$$ \\ $$$$\:\:\:\:\:\:{A}\:\:{Challanging}\:\:{Integral}: \\ $$$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{log}\left({x}\right).{log}\left(\mathrm{1}+{x}\right)}{\mathrm{1}−{x}}{dx} \\ $$$$ \\ $$$$ \\ $$

Question Number 143641 Answers: 1 Comments: 0

sin 160−sin 20=?

$$\mathrm{sin}\:\mathrm{160}−\mathrm{sin}\:\mathrm{20}=? \\ $$

Question Number 143638 Answers: 0 Comments: 2

......Calculus.... 𝛗:=^? ∫_0 ^( 1) ((ln(1−x)ln(x)(ln(((1−x)/x))))/x) dx m.n....

$$\:\:\:\:\:\:\:\:\:\:......{Calculus}.... \\ $$$$\boldsymbol{\phi}:\overset{?} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}\right){ln}\left({x}\right)\left({ln}\left(\frac{\mathrm{1}−{x}}{{x}}\right)\right)}{{x}}\:{dx} \\ $$$$\:\:{m}.{n}.... \\ $$$$ \\ $$

Question Number 143630 Answers: 1 Comments: 1

prove that if a and c are odd integers then ab+bc is even for every integer b?

$${prove}\:{that}\:{if}\:{a}\:{and}\:{c}\:{are}\:{odd}\:{integers} \\ $$$${then}\:{ab}+{bc}\:{is}\:{even}\:{for}\:{every}\:{integer}\:{b}? \\ $$

Question Number 143628 Answers: 1 Comments: 0

∫_x ^∝ t^(α−1) e^(it) dt=??

$$\int_{{x}} ^{\propto} {t}^{\alpha−\mathrm{1}} {e}^{{it}} {dt}=?? \\ $$

Question Number 143627 Answers: 2 Comments: 0

Question Number 143635 Answers: 1 Comments: 0

Question Number 143633 Answers: 1 Comments: 0

Question Number 143624 Answers: 1 Comments: 0

∫_0 ^( ∞) z^2 e^(1/z) dz

$$\int_{\mathrm{0}} ^{\:\infty} {z}^{\mathrm{2}} {e}^{\frac{\mathrm{1}}{{z}}} {dz} \\ $$

Question Number 143622 Answers: 1 Comments: 0

∫_0 ^∝ e^(2arctg(t^2 )) dt

$$\int_{\mathrm{0}} ^{\propto} {e}^{\mathrm{2}{arctg}\left({t}^{\mathrm{2}} \right)} {dt} \\ $$

Question Number 143794 Answers: 0 Comments: 3

Prove 3^(111) +1⋮223

$$\mathrm{Prove}\:\:\:\:\:\:\:\:\:\mathrm{3}^{\mathrm{111}} +\mathrm{1}\vdots\mathrm{223} \\ $$

Question Number 143609 Answers: 1 Comments: 0

Prove that:: Ω:=∫_0 ^( 1) ((ln^2 (1−x).ln(x))/x)dx=((−1)/2) ζ (4 ) Without using the “Beta function” m.n

$$\:\:\: \\ $$$$\:{Prove}\:{that}:: \\ $$$$\:\:\Omega:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right).{ln}\left({x}\right)}{{x}}{dx}=\frac{−\mathrm{1}}{\mathrm{2}}\:\zeta\:\left(\mathrm{4}\:\right) \\ $$$${Without}\:{using}\:{the}\:``{Beta}\:{function}'' \\ $$$$\:\:{m}.{n} \\ $$

Question Number 143606 Answers: 1 Comments: 0

Question Number 143603 Answers: 2 Comments: 0

.....Calculus..... Ω:=Σ_(n=1) ^∞ (1/(n^k (1+n))) (k≥ 2) ......

$$\:\:\:\:\:\:\:\:\:\:\:\:\:.....{Calculus}..... \\ $$$$\:\:\:\:\:\:\:\:\Omega:=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{k}} \left(\mathrm{1}+{n}\right)}\:\:\:\left({k}\geqslant\:\mathrm{2}\right)\:...... \\ $$

Question Number 143602 Answers: 2 Comments: 1

Question Number 143598 Answers: 3 Comments: 2

Question Number 143597 Answers: 0 Comments: 0

s = ut + (1/2)at^2 ⇒ t^2 + 2(u/a)t − 2(s/a) = 0 by the use of quadratic formula t = ((−((2u)/a) ± (√(((4u^2 )/a^2 ) + 4s)))/2) t = −(u/a) ± (√((u^2 /a^2 ) + s)) Victor Francis

$${s}\:=\:{ut}\:+\:\frac{\mathrm{1}}{\mathrm{2}}{at}^{\mathrm{2}} \:\:\Rightarrow\:\:{t}^{\mathrm{2}} \:+\:\mathrm{2}\frac{{u}}{{a}}{t}\:−\:\mathrm{2}\frac{{s}}{{a}}\:=\:\mathrm{0} \\ $$$${by}\:{the}\:{use}\:{of}\:{quadratic}\:{formula} \\ $$$${t}\:=\:\frac{−\frac{\mathrm{2}{u}}{{a}}\:\pm\:\sqrt{\frac{\mathrm{4}{u}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:+\:\mathrm{4}{s}}}{\mathrm{2}} \\ $$$${t}\:=\:−\frac{{u}}{{a}}\:\:\pm\:\:\sqrt{\frac{{u}^{\mathrm{2}} }{{a}^{\mathrm{2}} }\:+\:{s}} \\ $$$${Victor}\:\:{Francis} \\ $$$$ \\ $$$$ \\ $$

Question Number 143596 Answers: 1 Comments: 0

Given that the series (x/3^ )+(y/3^2 )+(x/3^3 )+(y/3^4 )+…+(x/3^(n−1) )+(y/3^n )=8, x,y∈R. Find the value of 3x+y.

$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{series} \\ $$$$\frac{{x}}{\mathrm{3}^{} }+\frac{{y}}{\mathrm{3}^{\mathrm{2}} }+\frac{{x}}{\mathrm{3}^{\mathrm{3}} }+\frac{{y}}{\mathrm{3}^{\mathrm{4}} }+\ldots+\frac{{x}}{\mathrm{3}^{{n}−\mathrm{1}} }+\frac{{y}}{\mathrm{3}^{{n}} }=\mathrm{8},\:{x},{y}\in\mathbb{R}. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{3}{x}+{y}. \\ $$

Question Number 143595 Answers: 2 Comments: 0

If f(x)=((10^x −10^(−x) )/(10^x +10^(−x) )), find f^( −1) (x).

$$\mathrm{If}\:{f}\left({x}\right)=\frac{\mathrm{10}^{{x}} −\mathrm{10}^{−{x}} }{\mathrm{10}^{{x}} +\mathrm{10}^{−{x}} },\:\mathrm{find}\:{f}^{\:−\mathrm{1}} \left({x}\right). \\ $$

Question Number 143593 Answers: 0 Comments: 0

soit (X_i ),i∈{1,2,......,n} une suite de variable aleartoire independante (iid) suivant la loi binomiale B(n,p) montrer que X_n ^_ converge en loi vers E(X) 2− montrer que (1/n)Σ_(i=1) ^2 X_(i ) ^2 converge en loi vers E (X^2 ) 3−montrer que (1/n)Σ_(i=1) ^n X_i Y_i converge en loi vers E(XY)

$${soit}\:\left(\boldsymbol{{X}}_{{i}} \right),{i}\in\left\{\mathrm{1},\mathrm{2},......,{n}\right\}\:{une}\:{suite}\:{de}\:{variable}\:{a}\boldsymbol{{leartoire}}\:\boldsymbol{{independante}} \\ $$$$\left(\boldsymbol{{iid}}\right)\:\boldsymbol{{suivant}}\:\boldsymbol{{la}}\:\boldsymbol{{loi}}\:\boldsymbol{{binomiale}}\:\boldsymbol{{B}}\left(\boldsymbol{{n}},\boldsymbol{{p}}\right)\: \\ $$$$\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\overset{\_} {\boldsymbol{{X}}}_{\boldsymbol{{n}}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\boldsymbol{{vers}}\:\boldsymbol{{E}}\left(\boldsymbol{{X}}\right) \\ $$$$\mathrm{2}−\:\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}} {\sum}}\boldsymbol{{X}}_{\boldsymbol{{i}}\:\:} ^{\mathrm{2}} \boldsymbol{{converge}}\:\boldsymbol{{en}}\:\boldsymbol{{loi}}\:\:\boldsymbol{{vers}}\:\:\:\:\:\:\:\:\boldsymbol{{E}}\:\left(\boldsymbol{{X}}^{\mathrm{2}} \right) \\ $$$$\mathrm{3}−\boldsymbol{{montrer}}\:\boldsymbol{{que}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\boldsymbol{{X}}_{{i}} \boldsymbol{{Y}}_{{i}} \:\boldsymbol{{converge}}\:\boldsymbol{{en}}\:\:\boldsymbol{{loi}}\:\boldsymbol{{vers}}\:\boldsymbol{{E}}\left(\boldsymbol{{XY}}\right) \\ $$

Question Number 143591 Answers: 1 Comments: 0

a;b;c>0 and a+b+c=k min((1/(1+a^2 )) + (1/(1+b^2 )) + (1/(1+c^2 )))=?

$${a};{b};{c}>\mathrm{0}\:\:{and}\:\:{a}+{b}+{c}={k} \\ $$$$\boldsymbol{{min}}\left(\frac{\mathrm{1}}{\mathrm{1}+{a}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}+{b}^{\mathrm{2}} }\:+\:\frac{\mathrm{1}}{\mathrm{1}+{c}^{\mathrm{2}} }\right)=? \\ $$

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