Saturday, 26 October 2019

calculus - Show that intinfty0fraclnxx4+1dx=fracpi2sqrt216



I could prove it using the residues but I'm interested to have it in a different way (for example using Gamma/Beta or any other functions) to show that
0ln(x)x4+1dx=π2216.



Thanks in advance.


Answer



One possible way is to introduce
I(s)=1160ys34dy1+y.
The integral you are looking for is obtained as I(0) after the change of variables y=x4.




Let us make in (1) another change of variables: t=y1+yy=t1t,dy=dt(1t)2. This gives
I(s)=11610t(t1t)s74dt(1t)2==11610ts34(1t)s14dt==116B(s+14,s+34)==116Γ(s+14)Γ(s+34)==π16sinπ(s+14).
Differentiating this with respect to s, we indeed get
I(0)=π2cosπ416sin2π4=π2216.



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