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Question Number 139342 Answers: 2 Comments: 1

Question Number 139332 Answers: 1 Comments: 0

Find the coefficient of x^(50) in the expression (1+x)^(1000) +2x(1+x)^(999) + 3x^2 (1+x)^(998) +...+1001x^(1000)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{50}} \:\mathrm{in} \\ $$$$\mathrm{the}\:\mathrm{expression}\:\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{1000}} \:+\mathrm{2x}\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{999}} + \\ $$$$\mathrm{3x}^{\mathrm{2}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{998}} +...+\mathrm{1001x}^{\mathrm{1000}} \\ $$

Question Number 139331 Answers: 2 Comments: 0

(√x)+(√y)=3 and (√(x+5))+(√(y+3))=5 find x and y

$$\sqrt{\boldsymbol{{x}}}+\sqrt{\boldsymbol{{y}}}=\mathrm{3}\:\:\boldsymbol{{and}} \\ $$$$\sqrt{\boldsymbol{{x}}+\mathrm{5}}+\sqrt{\boldsymbol{{y}}+\mathrm{3}}=\mathrm{5} \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{x}}\:\boldsymbol{{and}}\:\boldsymbol{{y}} \\ $$

Question Number 139328 Answers: 2 Comments: 1

Question Number 139323 Answers: 0 Comments: 0

Let f define such as f(1)=1,f(3)=3 ∀ n≥2 , f(2n)=f(n) f(4n+1)=2f(2n+1)−f(n) f(4n+3)=3f(2n+1)−2f(n) 1)Prove that ∀ n , f(n) is odd 2)Prove that if f(a_n )=a_n , then a_n =2^n −1 or a_n =2^n +1

$$\:{Let}\:{f}\:{define}\:{such}\:{as}\:\:{f}\left(\mathrm{1}\right)=\mathrm{1},{f}\left(\mathrm{3}\right)=\mathrm{3} \\ $$$$\forall\:{n}\geqslant\mathrm{2}\:\:,\:{f}\left(\mathrm{2}{n}\right)={f}\left({n}\right)\: \\ $$$${f}\left(\mathrm{4}{n}+\mathrm{1}\right)=\mathrm{2}{f}\left(\mathrm{2}{n}+\mathrm{1}\right)−{f}\left({n}\right) \\ $$$${f}\left(\mathrm{4}{n}+\mathrm{3}\right)=\mathrm{3}{f}\left(\mathrm{2}{n}+\mathrm{1}\right)−\mathrm{2}{f}\left({n}\right) \\ $$$$ \\ $$$$\left.\mathrm{1}\right){Prove}\:{that}\:\forall\:{n}\:,\:{f}\left({n}\right)\:{is}\:{odd} \\ $$$$\left.\mathrm{2}\right){Prove}\:{that}\:{if}\:\:\:{f}\left({a}_{{n}} \right)={a}_{{n}} \:, \\ $$$${then}\:\:{a}_{{n}} =\mathrm{2}^{{n}} −\mathrm{1}\:{or}\:{a}_{{n}} =\mathrm{2}^{{n}} +\mathrm{1} \\ $$

Question Number 139319 Answers: 0 Comments: 2

prove that are axactly 1729 positive integer solutions to the below equation 4x^4 +3y^3 +2z^2 +t=4311

$${prove}\:{that}\:{are}\:{axactly}\:\mathrm{1729}\:{positive} \\ $$$${integer}\:{solutions}\:{to}\:{the}\:{below}\:{equation} \\ $$$$\mathrm{4}{x}^{\mathrm{4}} +\mathrm{3}{y}^{\mathrm{3}} +\mathrm{2}{z}^{\mathrm{2}} +{t}=\mathrm{4311} \\ $$

Question Number 139371 Answers: 3 Comments: 0

by use Gamma function prove (1) ∫_0 ^( (π/8)) cos^3 4xdx=(1/6) (2) ∫_0 ^( π) sin^6 ((x/2))cos^8 ((x/2))dx=((5π)/2^(11) )

$${by}\:{use}\:{Gamma}\:{function}\:{prove}\: \\ $$$$ \\ $$$$\left(\mathrm{1}\right)\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{8}}} {cos}^{\mathrm{3}} \mathrm{4}{xdx}=\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$ \\ $$$$\left(\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\pi} {sin}^{\mathrm{6}} \left(\frac{{x}}{\mathrm{2}}\right){cos}^{\mathrm{8}} \left(\frac{{x}}{\mathrm{2}}\right){dx}=\frac{\mathrm{5}\pi}{\mathrm{2}^{\mathrm{11}} } \\ $$

Question Number 139313 Answers: 2 Comments: 0

∫_0 ^∞ (dx/((1+x^5 )^5 ))=((798(√2))/(5^5 (√(5−(√5)))))π

$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{5}} \right)^{\mathrm{5}} }=\frac{\mathrm{798}\sqrt{\mathrm{2}}}{\mathrm{5}^{\mathrm{5}} \sqrt{\mathrm{5}−\sqrt{\mathrm{5}}}}\pi \\ $$

Question Number 139296 Answers: 4 Comments: 0

∫_0 ^( 1) ((√x)/( (√(1−x))+(√x)))dx

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}+\sqrt{{x}}}{dx} \\ $$

Question Number 139290 Answers: 2 Comments: 0

∫_( 0) ^( ∞) ((sin(x^2 −arctan((1/x^2 ))))/( (√(1 + x^4 )))) dx

$$\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\:\frac{{sin}\left({x}^{\mathrm{2}} −{arctan}\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)\right)}{\:\sqrt{\mathrm{1}\:+\:{x}^{\mathrm{4}} }}\:{dx} \\ $$

Question Number 139288 Answers: 4 Comments: 0

IF ((x^3 +1)/(x^2 −1))=x+(√(6/x)) 6x^2 −5x+4=??

$$\boldsymbol{{IF}}\:\frac{\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{1}}{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{1}}=\boldsymbol{{x}}+\sqrt{\frac{\mathrm{6}}{\boldsymbol{{x}}}} \\ $$$$\mathrm{6}\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{5}\boldsymbol{{x}}+\mathrm{4}=?? \\ $$

Question Number 139286 Answers: 0 Comments: 1

(1/((1−x)^n ))=( ((n),(0) ))+( ((n),(1) ))x+( ((n),(2) ))x^2 +( ((n),(3) ))x^3 +...

$$\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{{n}} }=\left(\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}\right)+\left(\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\right){x}+\left(\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\right){x}^{\mathrm{2}} +\left(\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}\right){x}^{\mathrm{3}} +... \\ $$

Question Number 139285 Answers: 2 Comments: 0

∫_0 ^(π/2) sin^2 (x)cos^6 (x)dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}} \left({x}\right){cos}^{\mathrm{6}} \left({x}\right){dx} \\ $$

Question Number 139283 Answers: 2 Comments: 0

∫_0 ^∞ ((ln^2 x)/(x^2 +a^2 ))dx=(π/(8a))(π^2 +4ln^2 a) ,a>0

$$\int_{\mathrm{0}} ^{\infty} \frac{{ln}^{\mathrm{2}} {x}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{8}{a}}\left(\pi^{\mathrm{2}} +\mathrm{4}{ln}^{\mathrm{2}} {a}\right)\:\:\:\:\:\:\:\:\:,{a}>\mathrm{0} \\ $$

Question Number 139282 Answers: 1 Comments: 0

∫_0 ^∞ ((ln x)/(x^2 +a^2 ))dx=(π/(2a))∙ln a ,a>0

$$\int_{\mathrm{0}} ^{\infty} \frac{{ln}\:{x}}{{x}^{\mathrm{2}} +{a}^{\mathrm{2}} }{dx}=\frac{\pi}{\mathrm{2}{a}}\centerdot{ln}\:{a}\:\:\:\:\:\:\:\:\:\:,{a}>\mathrm{0}\: \\ $$

Question Number 139276 Answers: 1 Comments: 1

Question Number 139278 Answers: 2 Comments: 0

(√((3000000 + 3000000)/(4000000 + 4000000))) = ?

$$\sqrt{\frac{\mathrm{3000000}\:+\:\mathrm{3000000}}{\mathrm{4000000}\:+\:\mathrm{4000000}}}\:=\:? \\ $$

Question Number 139273 Answers: 0 Comments: 0

Question Number 139248 Answers: 2 Comments: 0

∫_a ^( b) f(t)g′(t)dt= ?

$$\int_{{a}} ^{\:{b}} {f}\left({t}\right){g}'\left({t}\right){dt}=\:? \\ $$

Question Number 139232 Answers: 1 Comments: 4

A man has ′n′ pair of black shoes and ′m′ pairs of brown shoes. Man hurriedly wear two shoes. What is probability that both of them are black?

$${A}\:{man}\:{has}\:'{n}'\:{pair}\:{of}\:{black}\:{shoes}\:{and}\:'{m}' \\ $$$${pairs}\:{of}\:{brown}\:{shoes}.\:{Man}\:{hurriedly}\:{wear} \\ $$$${two}\:{shoes}.\:{What}\:{is}\:{probability}\:{that}\:{both}\:{of} \\ $$$${them}\:{are}\:{black}? \\ $$

Question Number 139259 Answers: 1 Comments: 0

∫ ((cos x+(7)^(1/3) )/(sin x+(√6))) dx =?

$$\:\int\:\frac{\mathrm{cos}\:\mathrm{x}+\sqrt[{\mathrm{3}}]{\mathrm{7}}}{\mathrm{sin}\:\mathrm{x}+\sqrt{\mathrm{6}}}\:\mathrm{dx}\:=? \\ $$

Question Number 139257 Answers: 2 Comments: 0

Σ_(k=1) ^∞ (((−1)^k (2k−1))/k^2 ) =?

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

Question Number 139255 Answers: 2 Comments: 0

How do you find a point on the curve y = x^2 that is closest to the point? (16, 1/2)

$$ \\ $$How do you find a point on the curve y = x^2 that is closest to the point? (16, 1/2)

Question Number 139226 Answers: 0 Comments: 0

∫_0 ^1 e^x^2 dx=(e/(1+(2/(1+(4/(1+(6/(1+(8/(1+((10)/(1+...))))))))))))

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{{x}^{\mathrm{2}} } {dx}=\frac{{e}}{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{1}+\frac{\mathrm{4}}{\mathrm{1}+\frac{\mathrm{6}}{\mathrm{1}+\frac{\mathrm{8}}{\mathrm{1}+\frac{\mathrm{10}}{\mathrm{1}+...}}}}}} \\ $$

Question Number 139224 Answers: 2 Comments: 2

𝛗=∫_0 ^( 1) ((ln(1+x+x^2 ))/x)dx φ=∫_0 ^( 1) ((ln(1−x^3 )−ln(1−x))/x)dx =Ω+(π^2 /6) Ω=−Σ∫_0 ^( 1) (x^(3n−1) /n)dx=−Σ_(n=1 ) ^∞ (1/(3n^2 )) =((−π^2 )/(18)) .....≽ 𝛗=(π^2 /9) .......(π^2 /9)−(π^2 /(24)) =((5π^2 )/(72))

$$\:\:\:\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}+{x}+{x}^{\mathrm{2}} \right)}{{x}}{dx} \\ $$$$\:\:\:\phi=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}\left(\mathrm{1}−{x}^{\mathrm{3}} \right)−{ln}\left(\mathrm{1}−{x}\right)}{{x}}{dx} \\ $$$$\:\:\:\:=\Omega+\frac{\pi^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\:\:\:\:\Omega=−\Sigma\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{x}^{\mathrm{3}{n}−\mathrm{1}} }{{n}}{dx}=−\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{3}{n}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:=\frac{−\pi^{\mathrm{2}} }{\mathrm{18}}\:.....\succcurlyeq\:\boldsymbol{\phi}=\frac{\pi^{\mathrm{2}} }{\mathrm{9}} \\ $$$$\:\:\:\:\:\:\:.......\frac{\pi^{\mathrm{2}} }{\mathrm{9}}−\frac{\pi^{\mathrm{2}} }{\mathrm{24}}\:=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{72}}\: \\ $$

Question Number 139217 Answers: 1 Comments: 3

..... nice .... .... math.... prove that: 𝛗=∫_0 ^( (π/2)) ((ln(1+sin(x).cos(x)))/(tan(x)))dx=((5π^2 )/(72))

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:.....\:{nice}\:....\:....\:{math}.... \\ $$$$\:\:\:{prove}\:{that}: \\ $$$$\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{{ln}\left(\mathrm{1}+{sin}\left({x}\right).{cos}\left({x}\right)\right)}{{tan}\left({x}\right)}{dx}=\frac{\mathrm{5}\pi^{\mathrm{2}} }{\mathrm{72}} \\ $$

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