Monday, February 28, 2022

Integral of the day XXVII

An integral involving the Gamma function

Integral involving the Gamma functon Γ()


Today we show the proof of this integral posted by @Ali39342137 0(1+x4x2)nxk1dx=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2
This is a generalization for this other integral posted by @integralsbot with n=s and k=1: 0(1+x4x2)sdx=s2s+1π2Γ(s214)Γ(s2+14)
We previously posted another similar generalization here

Proof:

I=0(1+x4x2)nxk1dx=120(1+w2w)nwk21dw(wx2)=12k2+10tnk21(1t2)k21(t2+1)(t1+w2w)=12k2+1[0tnk2+1(1t2)k21dt+0tnk21(1t2)k21dt]=12k2+2[0sn2k4(1s)k21ds+0sn2k41(1s)k21ds](st2)=12k2+2[B(n2k4+1,k2)+B(n2k4,k2)]=12k2+2[Γ(n2k4+1)Γ(k2)Γ(n2+k4+1)+Γ(n2k4)Γ(k2)Γ(n2+k4)]=12k2+2[(n2k4)Γ(n2k4)Γ(k2)(n2+k4)Γ(n2+k4)+Γ(n2k4)Γ(k2)Γ(n2+k4)]=Γ(n2k4)Γ(k2)2k2+2Γ(n2+k4)[(n2k4)(n2+k4)+1]=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+k
0(1+x4x2)nxk1dx=Γ(n2k4)Γ(k2)2k2Γ(n2+k4)n2n+kn>k2

No comments:

Post a Comment

Series of the day

Series involving the digamma and the zeta functions The sum 1(n+1)pnq ...