Sunday, February 14, 2016

integration - Prove int10fracx1(x+1)logxtextdx=logfracpi2


Prove 10x1(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):=10xs1(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)=10xs1(x+1)logxdx=log(πΓ(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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