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let-consider-a-function-g-defined-by-g-a-0-1-dx-1-x-1-ax-Give-the-defined-Domain-of-g-and-simplify-g-




Question Number 67465 by ~ À ® @ 237 ~ last updated on 27/Aug/19
      let consider a function g defined by   g(a)=∫_0 ^1     (dx/( (√((1−x)(1+ax)))))    Give the defined Domain of g  and simplify g.
letconsiderafunctiongdefinedbyg(a)=01dx(1x)(1+ax)GivethedefinedDomainofgandsimplifyg.
Commented by ~ À ® @ 237 ~ last updated on 28/Aug/19
    D_g ={ a∈R/ g(a)∈R}  we know that  ∫_0 ^1  (dx/( (√(1−u^2 )) ))= (π/2) ( so  it converges). Then lim_(x→1)  (((√(1−u^2 )) )/( (√((1−u)(1+au))))) =(√(2/(1+a)))    by convergence theorem  g(a) exist if  (√((2/(1+a))  )) ∈ R  ⇔  a>−1  So D_g = ]−1;+∞[  If  a=0   g(0)=∫_0 ^1 (( dx)/( (√(1−x)) ))=[−2(√(1−x)) ]_0 ^1 =2  Now if  a≠0  and a∈]−1;+∞[  g(a)=∫_0 ^1      (du/( (√(u(1+a−au)))))   ( with  u=1−x )         = (1/( (√(1+a)))) ∫_0 ^1  u^((−1)/2) (1−((au)/(1+a)))^((−1)/2) du=(1/( (√(1+a)))) ∫_0 ^1  u^((1/2)−1) (1−u)^((3/2)−(1/2)−1) (1−(a/(1+a))u)^((−1)/2) du         = ((Γ((3/2)))/(Γ((1/2))Γ(1)(√(1+a)) )) . _2 F_1 ((1/2),(1/2),(3/2), (a/(1+a)))      because  ∣(a/(1+a))∣<1  using the formula  _2 F_1 ((1/2),(1/2),(3/2),z^2 )=((arcsinz)/z)    if   a∈]−1;0[     ,  (a/(1+a))<0     let  take z=i(√(∣(a/(1+a))∣))    g(a)=(1/(2(√(1+a)) )) . (1/(i(√(∣(a/(1+a))∣)) )) arcsin(i(√(∣(a/(1+a))∣)) )=_  ((arcsin(i(√((∣a∣)/(1+a))) ))/(2i(√(∣a∣)) ))  =((argsh((√(((∣a∣)/(1+a)) )) ))/(2(√(∣a∣)) ))     because  sin(x)=((e^(ix) −e^(−ix) )/(2i))= ((sh(ix))/i) ⇒ sin( arcsin(ix))=((sh(iarcsin(ix)))/i) ⇒ iargshx=arcsin(ix)  if  a>0   then  (a/(1+a))>0   we  take  z=(√(a/(1+a)))   g(a)=((arcsin((√((a/(1+a))  )) ))/(2(√a)))
Dg={aR/g(a)R}weknowthat01dx1u2=π2(soitconverges).Thenlimx11u2(1u)(1+au)=21+abyconvergencetheoremg(a)existif21+aRa>1SoDg=]1;+[Ifa=0g(0)=01dx1x=[21x]01=2Nowifa0anda]1;+[g(a)=01duu(1+aau)(withu=1x)=11+a01u12(1au1+a)12du=11+a01u121(1u)32121(1a1+au)12du=Γ(32)Γ(12)Γ(1)1+a.2F1(12,12,32,a1+a)becausea1+a∣<1usingtheformula2F1(12,12,32,z2)=arcsinzzifa]1;0[,a1+a<0lettakez=ia1+ag(a)=121+a.1ia1+aarcsin(ia1+a)=arcsin(ia1+a)2ia=argsh(a1+a)2abecausesin(x)=eixeix2i=sh(ix)isin(arcsin(ix))=sh(iarcsin(ix))iiargshx=arcsin(ix)ifa>0thena1+a>0wetakez=a1+ag(a)=arcsin(a1+a)2a
Commented by prof Abdo imad last updated on 28/Aug/19
at V(1)  g(a) ∼ ∫_0 ^1  (dx/( (√((1−x)(1+a)))))  g(a) exist ⇔ 1+a>0 ⇒a>−1 ⇒  D_g =]−1,+∞[ .
atV(1)g(a)01dx(1x)(1+a)g(a)exist1+a>0a>1Dg=]1,+[.
Commented by prof Abdo imad last updated on 28/Aug/19
2) we have g(a)=∫_0 ^1   (dx/( (√((1−x)(1+ax)))))  changrment (√(1−x))=t give 1−x =t^2  ⇒x=1−t^2   a>−1 ⇒ax>−x ⇒1+ax>1−x>0  g(a) =−∫_0 ^1      (((−2t)dt)/(t(√(1+a(1−t^2 )))))  =2 ∫_0 ^1   (dt/( (√(−at^2  +a+1))))  −at^2  +a+1 =0 ⇒at^2 −(a+1)=0  Δ^′  =0+a(a+1) =a^2  +a =a(a+1)  case −1< a<0 ⇒Δ^′ <0 ⇒no real roots  and  −at^2  +a+1=−a(t^2 −((a+1)/a)) and −((a+1)/a)>0 ⇒  we do the changement  t =(√(−((a+1)/a)))u ⇒  g(a) = 2 ∫_0 ^(√(−(a/(a+1))))        (1/( (√(−a))(√(−((a+1)/a)))(1+u^2 )))(√(−((a+1)/a)))du  =(2/( (√(−a)))) ∫_0 ^(√(−(a/(a+1))))      (du/(1+u^2 )) =(2/( (√(−a)))) arctan((√(−(a/(a+1)))))  ....be continued...
2)wehaveg(a)=01dx(1x)(1+ax)changrment1x=tgive1x=t2x=1t2a>1ax>x1+ax>1x>0g(a)=01(2\boldsymbolt)\boldsymboldt\boldsymbolt1+\boldsymbola(1\boldsymbolt2)=201dtat2+a+1at2+a+1=0at2(a+1)=0Δ=0+a(a+1)=a2+a=a(a+1)case1<a<0Δ<0norealrootsandat2+a+1=a(t2a+1a)anda+1a>0wedothechangementt=a+1aug(a)=20aa+11aa+1a(1+u2)a+1adu=2a0aa+1du1+u2=2aarctan(aa+1).becontinued
Commented by mathmax by abdo last updated on 28/Aug/19
case 2  if a>0  ⇒Δ^′ >0  ⇒t_1 =((√(a^2 +a))/a)   and t_2 =−((√(a^2 +a))/a)  g(a) = 2∫_0 ^1   (dt/( (√(−a(t−t_1 )(t−t_2 ))))) =(2/( (√a))) ∫_0 ^1    (dt/( (√((t_1 −t)(t−t_2 )))))  we use the changement  (√(t_1 −t))=u ⇒t_1 −t =u^2  ⇒t =t_1 −u^2   g(a) =(2/( (√a))) ∫_(√t_1 ) ^(√(t_1 −1))         (((−2u)du)/(u(√(t_1 −u^2 −t_2 )))) =((−4)/( (√a))) ∫_(√t_1 ) ^(√(t_1 −1))   (du/( (√(t_1 −t_2 −u^2 ))))  =((−4)/( (√a))) ∫_(√t_1 ) ^(√(t_1 −1))       (du/( (√(((√(t_1 −t_2 )))^2 −u^2 ))))  =_(u=(√(t_1 −t_2 ))z)       ((−4)/( (√a)))  ∫_((√t_1 )/( (√(t_1 −t_2 )))) ^((√(t_1 −1))/( (√(t_1 −t_2 ))))        (((√(t_1 −t_2 )) dz)/( (√(t_1 −t_2 ))((√(1−z^2 )))))  =((−4)/( (√a)))[arcsin(z)]_(√(t_1 /(t_1 −t_2 ))) ^(√((t_1 −1)/(t_1 −t_2 )))     =((−4)/( (√a))){ arcsin((√((t_1 −1)/(t_1 −t_2 ))))−arcsin((√(t_1 /(t_1 −t_2 ))))}
case2ifa>0Δ>0t1=a2+aaandt2=a2+aag(a)=201dta(tt1)(tt2)=2a01dt(t1t)(tt2)weusethechangementt1t=ut1t=u2t=t1u2g(a)=2at1t11(2u)duut1u2t2=4at1t11dut1t2u2=4at1t11du(t1t2)2u2=u=t1t2z4at1t1t2t11t1t2t1t2dzt1t2(1z2)=4a[arcsin(z)]t1t1t2t11t1t2=4a{arcsin(t11t1t2)arcsin(t1t1t2)}

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