Prove ∫10x−1(x+1)logxdx=logπ2
Tried contouring but couldn't get anywhere with a keyhole contour.
Geometric Series Expansion does not look very promising either.
Answer
Hint. One may set f(s):=∫10xs−1(x+1)logxdx,s>−1, then one is allowed to differentiate under the integral sign, getting f′(s)=∫10xsx+1dx=12ψ(s2+12)−12ψ(s2+1),s>−1,where we have used a standard integral representation of the digamma function.
One may recall that ψ:=Γ′/Γ, then integrating (2), observing that f(0)=0, one gets
f(s)=∫10xs−1(x+1)logxdx=log(√π⋅Γ(s2+1)Γ(s2+12)),s>−1,
from which one deduces the value of the initial integral by putting s:=1, recalling that Γ(12+1)=12Γ(12)=√π2.
Edit. The result (3) is more general than the given one.
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