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AllQuestion and Answers: Page 655

Question Number 143709    Answers: 1   Comments: 2

Question Number 143708    Answers: 3   Comments: 0

Prove that (3/4^ )+(3^2 /4^2 )+(3^3 /4^3 )+(3^4 /4^4 )+(3^5 /4^5 )+…=3

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{3}}{\mathrm{4}^{} }+\frac{\mathrm{3}^{\mathrm{2}} }{\mathrm{4}^{\mathrm{2}} }+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{4}^{\mathrm{3}} }+\frac{\mathrm{3}^{\mathrm{4}} }{\mathrm{4}^{\mathrm{4}} }+\frac{\mathrm{3}^{\mathrm{5}} }{\mathrm{4}^{\mathrm{5}} }+\ldots=\mathrm{3} \\ $$

Question Number 143706    Answers: 1   Comments: 0

2^x +9^y =x^2 +9xy+y^2 Find x,y∈N

$$\mathrm{2}^{\mathrm{x}} +\mathrm{9}^{\mathrm{y}} =\mathrm{x}^{\mathrm{2}} +\mathrm{9xy}+\mathrm{y}^{\mathrm{2}} \\ $$$$\mathrm{Find}\:\mathrm{x},\mathrm{y}\in\mathbb{N} \\ $$

Question Number 143702    Answers: 2   Comments: 0

n ∈ IN. I_n = ∫_1 ^( e) x^(n+1) lnx dx. 1. prove that (I_n ) is positive and increasing. 2. using a part−by−part integration, calculate I_n .

$${n}\:\in\:\mathrm{IN}. \\ $$$${I}_{{n}} \:=\:\int_{\mathrm{1}} ^{\:\mathrm{e}} {x}^{{n}+\mathrm{1}} {lnx}\:{dx}. \\ $$$$\mathrm{1}.\:\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\:\left(\boldsymbol{{I}}_{\boldsymbol{{n}}} \right)\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{increasing}}. \\ $$$$\mathrm{2}.\:\boldsymbol{\mathrm{using}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{part}}−\boldsymbol{\mathrm{by}}−\boldsymbol{\mathrm{part}}\:\boldsymbol{\mathrm{integration}},\:\boldsymbol{\mathrm{calculate}}\:\boldsymbol{{I}}_{\boldsymbol{{n}}} . \\ $$

Question Number 143701    Answers: 1   Comments: 1

Question Number 143698    Answers: 0   Comments: 5

My first Contribution to this forum. One year later Q 98831

$$\:{My}\:{first}\:{Contribution}\:{to}\:{this}\:{forum}.\: \\ $$$${One}\:{year}\:{later}\: \\ $$$${Q}\:\mathrm{98831} \\ $$

Question Number 143688    Answers: 0   Comments: 2

Question Number 143684    Answers: 1   Comments: 0

x^3 +x−1=^3 (√(2x^3 +11))+(√(5x^2 +16)) Find x∈R

$$\mathrm{x}^{\mathrm{3}} +\mathrm{x}−\mathrm{1}=^{\mathrm{3}} \sqrt{\mathrm{2x}^{\mathrm{3}} +\mathrm{11}}+\sqrt{\mathrm{5x}^{\mathrm{2}} +\mathrm{16}} \\ $$$$\mathrm{Find}\:\mathrm{x}\in\mathbb{R} \\ $$

Question Number 143680    Answers: 1   Comments: 1

tan 76=4 sin^2 14=?

$$\mathrm{tan}\:\mathrm{76}=\mathrm{4} \\ $$$$\mathrm{sin}\:^{\mathrm{2}} \mathrm{14}=? \\ $$

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

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

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