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[tex]I_n(x)=\int \frac{n!dx}{x(x+1)\cdots (x+n)}[/tex]

The desired result is an evaluation of the sum

[tex]S_n(x)=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{c}n\\k\end{array}\right) (x+k)\log (x+k)[/tex]

and so it is important that the evaluation of [tex]I_n(x)[/tex] be not by means of partial fraction decomposition, for this is already of note that [tex]S_n(x)[/tex] may be evaluated, that is to say: it is known to me that

[tex]\int\frac{n!dx}{x(x+1)\cdots (x+n)}=\int\sum_{k=0}^{n}\left(\begin{array}{c}n\\k\end{array}\right) \frac{(-1)^k}{x+k}dx = \sum_{k=0}^{n}\left(\begin{array}{c}n\\k\end{array}\right) (-1)^{k}\log (x+k)+C[/tex]

whence integrating once again will produce an evaluation for [tex]S_n(x)[/tex].

I was working on developing a recurrence relation for [tex]I_n(x)[/tex] that I may solve it via generating functions; here of note is that it is also known to me that

[tex]I_{n+1}(x)=I_n(x)-I_n(x+1)[/tex]

yet this is not useful to that end. What is desired is a non-homogeneous recurrence. Any thoughts?

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# Homework Help: A challenging Integral

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