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=−π2√216.
Thanks in advance.
Answer
One possible way is to introduce
I(s)=116∫∞0ys−34dy1+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+y⟺y=t1−t,dy=dt(1−t)2. This gives
I(s)=116∫10t⋅(t1−t)s−74⋅dt(1−t)2==116∫10ts−34(1−t)−s−14dt==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=−π2√216.
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