How to calculate $\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$

Question

I met an integral $$\int_0^1\cdots\int_0^1\frac{1}{x_1+x_2+\cdots+x_n+1}dx_1\cdots dx_n$$ I calculated $n=1,2,3$ and made an induction! then i got the result: $$\frac{1}{(n-1)!}\sum_{k=1}^{n}(-1)^{n-k}\binom{n}{k}(1+k)^{n-1}\ln(1+k)$$

but how could i get the result without induction? who can help ,thanks!

Answer 1

Your result follows from the fact that the sum of $n$ independent, uniformly distributed random variables over $[0,1]$, has a well-known characteristic function. That fact follows from Fourier inversion.