n-fold integral

Per

Nov 2009
10
0
Can anyone give me any suggestion of how to solve this integral? Or better - has anyone seen any solution to this kind of integral?


\(\displaystyle
\int_0^1\int_0^1\cdots \int_0^1\frac{1}{\left( c+x_{1}+x_{2}+\cdots +x_{n}\right) }dx_{1}\cdots dx_{n}
\)

I would be grateful for any help!
 
Oct 2009
4,261
1,836
Can anyone give me any suggestion of how to solve this integral? Or better - has anyone seen any solution to this kind of integral?


\(\displaystyle
\int_0^1\int_0^1\cdots \int_0^1\frac{1}{\left( c+x_{1}+x_{2}+\cdots +x_{n}\right) }dx_{1}\cdots dx_{n}
\)

I would be grateful for any help!

Assuming \(\displaystyle c+x_1+\ldots+x_n>0\,,\,\,\forall \,0\leq x_i\leq 1\) :

Beginning with the first one: \(\displaystyle \int^1_0\frac{1}{c+x_1+\ldots +x_n}\,dx_n=[\ln(c+x_1+\ldots +x_n)]^1_0=\) \(\displaystyle \ln(c+x_1+\ldots +x_{n-1}+1)-\ln(c+x_1+\ldots +x_{n-1})\) ... I purposedly

didn't write this as the logarithm of a quotient since for the next step is easier to integrate this way.

Next step: \(\displaystyle \int^1_0\left(\ln(c+x_1+\ldots +x_{n-1}+1)-\ln(c+x_1+\ldots +x_{n-1})\right)\,dx_{n-1}=\) \(\displaystyle \left[(c+x_1+\ldots +x_{n-1}+1)(\ln(c+x_1+\ldots +x_{n-1}+1)-1)\right]^1_0\)

\(\displaystyle -\left[(c+x_1+\ldots +x_{n-1})(\ln(c+x_1+\ldots +x_{n-1})-1)\right]^1_0=\) ...etc.

The above doesn't look too hard but it looks pretty messy and ugly...I don't know if there's a nice expression for it all.

Tonio