$$\int \frac{dx}{x(x+p)(x+2p)(x+3p)...(x+(n-1)p)}= ?$$
I'm trying to solve this integral, and as I usually do in these cases, I break the expression into partial fractions, but I find this case somehow tough, despite the thing that all the factors are liniar, I have no idea on how to find just a bunch of constants. Any ideas? Thanks!
I have no idea to calculate it directly, but I guessed and easily verified my answer $$\int \frac{dx}{x(x+p)(x+2p)(x+3p)...(x+(n-1)p)}= \\ \dfrac{1}{\Gamma(n)}\cdot\sum_{k=1}^{n} \binom{n-1}{k-1}\ln|x+k-1|(-1)^{k-1}+C$$