# How could we express the above integral in a similar form if the upper limit of the integral was some constant T rather than oo?

Try to deal with a problem, that might actually be related to Laplace transform. Here is brief overview. Let $P\left(t\right)={p}_{m}{t}^{m}+\cdots +{p}_{1}t+{p}_{0}$. Then we know that,it is possible to express the following integral in the form.
${\int }_{0}^{\mathrm{\infty }}{e}^{-zt}P\left(t\right)=\sum _{a=0}^{m}{p}_{a}\frac{a!}{{z}^{a+1}},$
where the equality comes from the property of Laplace transform.
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Jean Deleon
You can use the lower incomplete gamma function

So that

If you take $T\to \mathrm{\infty }$ you have $\gamma \left(k+1,zT\right)\to \mathrm{\Gamma }\left(k+1\right)=k!$ which leads back to your last expression