Integral of binomial coefficients Let the integral in question be given by
\begin{align}
f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt.
\end{align}
The integral can also be seen in the form
\begin{align}
f_{n}(x) = \frac{1}{n!} \, \int_{1}^{x} \frac{\Gamma(t) \, dt}{\Gamma(t-n)} = \frac{(-1)^{n+1}}{n!} \int_{1}^{1-x} (u)_{n} \, du.
\end{align}
The first few values are
\begin{align}
f_{0}(x) &= x-1 \\
f_{1}(x) &= \frac{(x-1)^{2}}{2!} \\
f_{2}(x) &= \frac{2}{4!} \, (2 x^{3} - 9 x^{2} + 12 x - 5).
\end{align}
The question then becomes:


*

*What is the general series form of $f_{n}(x)$ ?

*Is there a connection to known special functions? In particular can $f_{n}(x)$ be related to the Bernoulli numbers of the second kind? 



Here's another variation of the theme based upon Stirling Numbers.
Starting from
  \begin{align*}
  f_n(x)&=\int_1^x\binom{t-1}{n}dt
  =\frac{1}{n!}\int_1^x{(t-1)}_ndt
  =\frac{1}{n!}\int_0^{x-1}{(u)}_ndu
  \end{align*}
  we can use the Stirling Numbers of the first kind $s(n,k)$ which can be defined for $n\geq 0$ and $0\leq k \leq n$ by
  \begin{align*}
  {(u)}_n=\sum_{k=0}^ns(n,k)u^k\tag{1}
  \end{align*}

Using this relationship we obtain for $n\geq 0$
\begin{align*}
  f_n(x)&=\frac{1}{n!}\int_0^{x-1}\sum_{k=0}^ns(n,k)u^kdu\\
  &=\frac{1}{n!}\sum_{k=0}^ns(n,k)\int_0^{x-1}u^kdu\\
  &=\frac{1}{n!}\sum_{k=0}^ns(n,k)\left.\frac{1}{k+1}u^{k+1}\right|_0^{x-1}\\
  &=\frac{1}{n!}\sum_{k=0}^ns(n,k)\frac{1}{k+1}(x-1)^{k+1}
  \end{align*}

We conclude
\begin{align*}
f_n(x)=
\frac{1}{n!}\sum_{k=0}^n\frac{s(n,k)}{k+1}(x-1)^{k+1}&\qquad n\geq 0\tag{2}\\
\end{align*}

An exponential generating function  $\sum_{n=0}^{\infty}f_n(x)\frac{z^n}{n!}$ can be derived from the generating function of the Stirling Numbers of the first kind.

Generating function: In accordance to the answer of @SylvainL. we claim
The following is valid
\begin{align*}
\sum_{n=0}^{\infty}f_n(x)\frac{z^n}{n!}=\frac{(1+z)^{x-1}-1}{\log(1+z)}
\end{align*}

We obtain from (1)
\begin{align*}
(1+z)^u&=\sum_{n=0}^{\infty}\binom{u}{n}z^n\\
&=\sum_{n=0}^{\infty}{(u)}_n\frac{z^n}{n!}\\
&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^ns(n,k)u^k\right)\frac{z^n}{n!}\tag{3}
\end{align*}

In the following we consider $f_n(x+1)$ instead of $f_n(x)$.
We observe
\begin{align*}\sum_{n=0}^{\infty}f_n(x+1)\frac{z^n}{n!}
&=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}\frac{s(n,k)}{k+1}x^{k+1}\right)\frac{z^n}{n!}\tag{4}\\
&=\int_0^x\sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}s(n,k)u^{k}\right)\frac{z^n}{n!}du\\
&=\int_0^{x}(1+z)^udu\tag{5}\\
&=\int_0^xe^{u\log(1+z)}du\\
&=\left.\frac{1}{\log(1+z)}e^{u\log(1+z)}\right|_0^x\\
&=\frac{(1+z)^x-1}{\log(1+z)}\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Box
\end{align*}
  and the claim follows.

Comment:


*

*In (4) we use the representation (2) of $f_n(x)$

*In (5) we use the generating function of $s(n,k)$ from (3)

Calculation for small values $n=0,1,2$:
Since $$s(0,0)=1,s(1,1)=1,s(2,1)=-1,s(2,2)=1$$
  we obtain
  \begin{align*}
f_0(x)&=s(0,0)(x-1)=x-1\\
  f_1(x)&=s(1,1)\frac{1}{2}(x-1)^2=\frac{1}{2}(x-1)^2\\
  f_2(x)&=\frac{1}{2}\left(s(2,1)\frac{1}{2}(x-1)^2+s(2,2)\frac{1}{3}(x-1)^3\right)\\
  &=-\frac{1}{4}(x-1)^2+\frac{1}{6}(x-1)^3\\
  &=\frac{1}{12}(2x^3-9x^2+12x-5)
  \end{align*}

Note: We don't need analytical considerations for the calculations above, since we can state that all calculations are done within the ring of formal power series.
