Sunday, 2 December 2018

calculus - Closed-form of int10xnoperatornameli(xm),dx



I've conjectured, that for n0 and m1 integers
10xnli(xm)dx?=1n+1ln(m+n+1m),


where li is the logarithmic integral.



Although there is a known antiderivative of xnli(xm), the simplification of the expression seems not trivial. I think there are other ways to evaluate this definite integral problem.



How could we prove this identity?


Answer



10xnli(xm)dx=1m10z(n+1)/m1li(z)dz


but integration by parts gives:

10zα1li(z)dz=zα1αli(z)|1010zα1αlogzdz

and the last integral can be computed through the substitution z=et and Frullani's theorem:



10zα1log(z)dz=log(α+1)


hence:
10xnli(xm)dx=1n+1log(n+1m+1)

as you claimed.


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