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Question Number 173978 Answers: 1 Comments: 0

Solve system of equations: x+((3x−y)/(x^2 +y^2 ))=3 y−((x+3y)/(x^2 +y^2 ))=0

$$\mathrm{Solve}\:\mathrm{system}\:\mathrm{of}\:\mathrm{equations}: \\ $$$$\mathrm{x}+\frac{\mathrm{3x}−\mathrm{y}}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }=\mathrm{3} \\ $$$$\mathrm{y}−\frac{\mathrm{x}+\mathrm{3y}}{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }=\mathrm{0} \\ $$$$ \\ $$

Question Number 173976 Answers: 1 Comments: 0

B(a,b)=∫_0 ^1 x^(a−1) (1−x)^(b−1) dx Γ(s)= ∫_0 ^∞ t^(s−1) e^(−t) dt Why B(a,b)= ((Γ(a)Γ(b))/(Γ(a+b))) ?

$$ \\ $$$$\:\:\:\:{B}\left({a},{b}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{a}−\mathrm{1}} \left(\mathrm{1}−{x}\right)^{{b}−\mathrm{1}} {dx}\: \\ $$$$\:\:\:\:\:\:\:\Gamma\left({s}\right)=\:\int_{\mathrm{0}} ^{\infty} {t}^{{s}−\mathrm{1}} {e}^{−{t}} {dt} \\ $$$$ \\ $$$$\:\:{Why}\:\:\:\:{B}\left({a},{b}\right)=\:\frac{\Gamma\left({a}\right)\Gamma\left({b}\right)}{\Gamma\left({a}+{b}\right)}\:? \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 173972 Answers: 0 Comments: 0

Question Number 173971 Answers: 0 Comments: 2

Question Number 173970 Answers: 1 Comments: 3

Question Number 173975 Answers: 0 Comments: 0

∫_0 ^∞ (y^(a−1) /((1+y)^b )) dy =^(u=(1/(1+y))) ∫_0 ^1 (((1−u)^(a−1) )/u^(a−1) ) u^b (du/u^2 ) = ∫_0 ^1 u^(b−a−1) (1−u)^(a−1) du = B(b−a,a)=((Γ(b−a)Γ(a))/(Γ(b)))

$$ \\ $$$$\: \\ $$$$ \\ $$$$\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \frac{{y}^{{a}−\mathrm{1}} }{\left(\mathrm{1}+{y}\right)^{{b}} }\:{dy}\:\overset{{u}=\frac{\mathrm{1}}{\mathrm{1}+{y}}} {=}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left(\mathrm{1}−{u}\right)^{{a}−\mathrm{1}} }{{u}^{{a}−\mathrm{1}} }\:{u}^{{b}} \:\:\frac{{du}}{{u}^{\mathrm{2}} }\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{u}^{{b}−{a}−\mathrm{1}} \left(\mathrm{1}−{u}\right)^{{a}−\mathrm{1}} {du} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{B}\left({b}−{a},{a}\right)=\frac{\Gamma\left({b}−{a}\right)\Gamma\left({a}\right)}{\Gamma\left({b}\right)} \\ $$$$\:\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

Question Number 173948 Answers: 1 Comments: 4

Question Number 173946 Answers: 1 Comments: 0

If 3sin(x)+4cos(y)−3sin(y)=8 then .. sin(x) + sin(y)=?

$$ \\ $$$$\:\:\:{If}\:\mathrm{3}{sin}\left({x}\right)+\mathrm{4}{cos}\left({y}\right)−\mathrm{3}{sin}\left({y}\right)=\mathrm{8} \\ $$$$\:\:\:{then}\:..\:\:\:\:{sin}\left({x}\right)\:+\:{sin}\left({y}\right)=? \\ $$$$ \\ $$$$ \\ $$

Question Number 173933 Answers: 1 Comments: 1

Question Number 173958 Answers: 0 Comments: 0

Question Number 173926 Answers: 2 Comments: 3

Q: a_( n ) is an arithmatic sequence. a ( first term) and d (difference ) such that , a_( a) + a_( d) = a_( ad) find : a_( n) =? note: a , d ∈ N

$$ \\ $$$$\:\:\:\:\:{Q}: \\ $$$$\:\:\:\:\:\:\:{a}_{\:{n}\:} \:\:{is}\:{an}\:{arithmatic}\:{sequence}. \\ $$$$\:\:\:\:\:\:\:\:\:\:{a}\:\left(\:{first}\:{term}\right)\:{and}\:\:{d}\:\left({difference}\:\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{such}\:{that}\:,\:\:{a}_{\:{a}} \:+\:{a}_{\:{d}} \:=\:{a}_{\:{ad}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:{find}\:\:\::\:\:\:\:{a}_{\:{n}} \:\:=? \\ $$$$\:\:\:\:\:\:\:\:\:\:{note}:\:\:{a}\:\:,\:\:{d}\:\:\:\in\:\mathbb{N}\:\: \\ $$$$\:\:\:\:\:\:\: \\ $$

Question Number 173923 Answers: 1 Comments: 1

Question Number 173917 Answers: 0 Comments: 2

prove: (−1)!

$${prove}: \\ $$$$\left(−\mathrm{1}\right)! \\ $$

Question Number 173908 Answers: 0 Comments: 6

find the value of b so that the line y=b divides the region bound by the graphs of the two functinos , into two regions of equal area. f(x)=9−x^2 and g(x)=0

$${find}\:{the}\:{value}\:{of}\:{b}\:{so}\:{that}\:{the}\:{line}\:{y}={b} \\ $$$${divides}\:{the}\:{region}\:{bound}\:{by}\:{the}\:{graphs}\:{of} \\ $$$${the}\:{two}\:{functinos}\:,\:{into}\:{two}\:{regions}\:{of}\:{equal} \\ $$$${area}. \\ $$$${f}\left({x}\right)=\mathrm{9}−{x}^{\mathrm{2}} \:{and}\:{g}\left({x}\right)=\mathrm{0} \\ $$

Question Number 173956 Answers: 0 Comments: 1

Question Number 173909 Answers: 0 Comments: 2

Ω=∫_0 ^( 1) (( ln^( 2) (1−x))/((1+x )^( 2) )) = Li_2 ((1/2) ) −−− Solution −−− Ω =^(i.b.p) {[−(1/(1+x))ln^( 2) (1−x)]_0 ^1 −2∫_0 ^( 1) ((ln(1−x))/((1−x)(1+x)))dx =lim_( x→1^− ) −(1/(1+x)) ln^( 2) (1−x)−∫_0 ^( 1) ((ln(1−x))/(1−x)) +((ln(1−x))/(1+x))dx = lim_( x→1^− ) (1/2) ln^( 2) (1−x)−(1/(1+x))ln^( 2) (1−x)−Φ =lim_( x→1^( −) ) (((x−1)/(2(1+x))))ln^( 2) (1−x)−Φ = −Φ =_(derived) ^(earlier) −(−(π^( 2) /(12)) +(1/2)ln^( 2) (2)) ⇒ Ω = (π^( 2) /(12)) −(1/2) ln^( 2) (2 )=Li_2 ((1/2))

Question Number 173902 Answers: 1 Comments: 0

Q: How many common three−digit numbers are there in the following two sequences? { (( a_n = 1 , 5 , 9 ,13 , ...)),(( b_( m) = 4 , 7 , 10 , 13 ,...)) :}

$$ \\ $$$$\:\:{Q}:\:{How}\:{many}\:{common}\:{three}−{digit}\:{numbers} \\ $$$$\:\:\:\:{are}\:{there}\:{in}\:{the}\:{following} \\ $$$$\:\:\:\:\:{two}\:{sequences}? \\ $$$$\:\:\:\:\:\begin{cases}{\:\:{a}_{{n}} \:=\:\mathrm{1}\:\:,\:\mathrm{5}\:,\:\mathrm{9}\:,\mathrm{13}\:,\:...}\\{\:\:{b}_{\:{m}} \:=\:\mathrm{4}\:,\:\mathrm{7}\:,\:\mathrm{10}\:,\:\mathrm{13}\:,...}\end{cases} \\ $$

Question Number 173893 Answers: 1 Comments: 1

if ∫(x) = sinx than prove that, {∫(x)^4 } + {∫(x)}^2 = 1

$$\mathrm{if}\:\int\left(\mathrm{x}\right)\:=\:\mathrm{sinx}\:\mathrm{than}\:\mathrm{prove}\:\mathrm{that}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left\{\int\left(\mathrm{x}\right)^{\mathrm{4}} \right\}\:+\:\left\{\int\left(\mathrm{x}\right)\right\}^{\mathrm{2}} \:=\:\mathrm{1} \\ $$

Question Number 173894 Answers: 2 Comments: 2

If secA − tanA = Q than prove that, cosecA = ((1 + Q^2 )/(1 − Q^2 ))

$$\mathrm{If}\:\:\mathrm{secA}\:−\:\mathrm{tanA}\:=\:\mathrm{Q}\:\mathrm{than}\:\mathrm{prove}\:\mathrm{that},\: \\ $$$$\:\:\:\:\:\:\:\mathrm{cosecA}\:=\:\frac{\mathrm{1}\:+\:\mathrm{Q}^{\mathrm{2}} }{\mathrm{1}\:−\:\mathrm{Q}^{\mathrm{2}} }\: \\ $$

Question Number 173889 Answers: 0 Comments: 1

the anser is the folowing

$${the}\:{anser}\:{is}\:{the}\:{folowing} \\ $$

Question Number 173887 Answers: 0 Comments: 0