I've conjectured, that for n≥0 and m≥1 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=1m∫10z(n+1)/m−1li(z)dz
but integration by parts gives:
∫10zα−1li(z)dz=zα−1αli(z)|10−∫10zα−1αlogzdz
and the last integral can be computed through the substitution z=e−t and Frullani's theorem:
∫10zα−1log(z)dz=log(α+1)
hence:
∫10xnli(xm)dx=−1n+1log(n+1m+1)
as you claimed.
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