Evaluate $\int_a^s\frac{(t-s)^n}{t(t-z)^{n+1}}dt.$ Evaluate: $$\int_a^s \frac{(t-s)^n}{t(t-z)^{n+1}}dt,$$ where $n>0$ is an integer, $0<a<s$. I'm not sure what values $z$ should take yet, but any help with the integral would be helpful.
I believe this should be expressible as a finite sum, but am having difficulty getting a hold on that.
EDIT - I think one possible approach might be to use $y=t-z$ in the original integral so that we have a factor of $1/((y+z)y^{n+1})$. That way we can use partial fractions to obtain $$\frac{1}{(y+z)y^{n+1}}=\frac{(-1)^{n+1}}{z^{n+1}(y+z)}+\sum_{k=0}^n\frac{(-1)^k}{z^{k+1}y^{n-k+1}},$$ and go from there.
 A: Case $1$: $z=0$
Then $\int_a^s\dfrac{(t-s)^n}{t^{n+2}}dt$
$=\int_a^s\dfrac{\sum\limits_{k=0}^nC_k^n(-1)^ks^kt^{n-k}}{t^{n+2}}dt$
$=\int_a^s\sum\limits_{k=0}^n\dfrac{(-1)^kn!s^kt^{-k-2}}{k!(n-k)!}dt$
$=\left[-\sum\limits_{k=0}^n\dfrac{(-1)^kn!s^k}{(k+1)!(n-k)!t^{k+1}}\right]_a^s$
$=\sum\limits_{k=0}^n\dfrac{(-1)^kn!s^k}{(k+1)!(n-k)!a^{k+1}}-\sum\limits_{k=0}^n\dfrac{(-1)^kn!}{(k+1)!(n-k)!s}$
Case $2$: $z\neq0$
With reference to http://www.math-cs.gordon.edu/courses/ma225/handouts/heavyside.pdf,
Then $\int_a^s\dfrac{(t-s)^n}{t(t-z)^{n+1}}dt$
$=\int_a^s\left(\dfrac{(t-s)^n}{t(t-z)^{n+1}}+\dfrac{s^n}{z^{n+1}t}\right)dt-\int_a^s\dfrac{s^n}{z^{n+1}t}dt$
$=\int_a^s\left(\dfrac{(t-s)^n}{t(t-z)^{n+1}}+\dfrac{s^n(t-z)^{n+1}}{z^{n+1}t(t-z)^{n+1}}\right)dt-\left[\dfrac{s^n\ln t}{z^{n+1}}\right]_a^s$
$=\int_a^s\left(\dfrac{\sum\limits_{m=0}^nC_m^n(-1)^{n-m}s^{n-m}t^m}{t(t-z)^{n+1}}+\dfrac{s^n\sum\limits_{m=0}^{n+1}C_m^{n+1}(-1)^{n-m+1}z^{n-m+1}t^m}{z^{n+1}t(t-z)^{n+1}}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
$=\int_a^s\left(\sum\limits_{m=1}^n\dfrac{(-1)^{n-m}n!s^{n-m}t^m}{m!(n-m)!t(t-z)^{n+1}}+\sum\limits_{m=1}^{n+1}\dfrac{(-1)^{n-m+1}(n+1)!s^nz^{-m}t^m}{m!(n-m+1)!t(t-z)^{n+1}}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
$=\int_a^s\left(\sum\limits_{m=1}^n\dfrac{(-1)^{n-m}n!s^{n-m}t^{m-1}}{m!(n-m)!(t-z)^{n+1}}+\sum\limits_{m=1}^{n+1}\dfrac{(-1)^{n-m+1}(n+1)!s^nz^{-m}t^{m-1}}{m!(n-m+1)!(t-z)^{n+1}}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
$=\int_a^s\left(\sum\limits_{m=1}^n\dfrac{(-1)^{n-m}n!s^{n-m}(t-z+z)^{m-1}}{m!(n-m)!(t-z)^{n+1}}+\sum\limits_{m=1}^{n+1}\dfrac{(-1)^{n-m+1}(n+1)!s^nz^{-m}(t-z+z)^{m-1}}{m!(n-m+1)!(t-z)^{n+1}}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
$=\int_a^s\left(\sum\limits_{m=1}^n\sum\limits_{k=0}^m\dfrac{(-1)^{n-m}n!s^{n-m}C_k^mz^{m-k-1}(t-z)^k}{m!(n-m)!(t-z)^{n+1}}+\sum\limits_{m=1}^{n+1}\sum\limits_{k=0}^m\dfrac{(-1)^{n-m+1}(n+1)!s^nz^{-m}C_k^mz^{m-k-1}(t-z)^k}{m!(n-m+1)!(t-z)^{n+1}}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
$=\int_a^s\left(\sum\limits_{m=1}^n\sum\limits_{k=0}^m\dfrac{(-1)^{n-m}n!s^{n-m}z^{m-k-1}(t-z)^{k-n-1}}{k!(n-m)!(m-k)!}+\sum\limits_{m=1}^{n+1}\sum\limits_{k=0}^m\dfrac{(-1)^{n-m+1}(n+1)!s^nz^{-k-1}(t-z)^{k-n-1}}{k!(n-m+1)!(m-k)!}\right)dt-\dfrac{s^n}{z^{n+1}}\ln\dfrac{s}{a}$
