# Simplifying Multiple Integral Expressions

Suppose i have the following multiple integral expressions...

\begin{align} I_1 = {} & \int_x^{x+1}x_1^n\,dx_1 \\[10pt] I_2 = {} & \int_x^{x+1}\int_0^{x_1}x_2^n\,dx_2\,dx_1-\int_0^xx_1^n\,dx_1 \\[10pt] I_3 = {} & \int_x^{x+1}\int_0^{x_1}\int_0^{x_2}x_3^n\,dx_3\,dx_2\,dx_1-\int_0^x \int_0^{x_1} x_2^n\,dx_2\,dx_1-\frac{1}{2}\int_0^xx_1^n\,dx_1 \\[10pt] I_4 = {} & \int_x^{x+1}\int_0^{x_1} \int_0^{x_2} \int_0^{x_3} x_4^n \,dx_4 \,dx_3 \,dx_2\,dx_1 -\int_0^x\int_0^{x_1} \int_0^{x_2} x_3^n \,dx_3\,dx_2\,dx_1 \\[5pt] & {} -\frac{1}{2} \int_0^x \int_0^{x_1} x_2^n \, dx_2 \, dx_1-\frac{1}{6}\int_0^xx_1^n \, dx_1 \end{align}

Is there a way to generalize these expressions for higher and higher cases of $I_k$? There is clearly a pattern, but writing this out for $I_k, k>4$ seems daunting and I wasn't sure if there was an agreed upon way to 'compactify' these types of expressions.

• Have you tried to integrate the expressions? Commented Nov 2, 2017 at 19:34

For $k \in \mathbb{N}$,$$I_k=n! \left(\frac{(x+1)^{n-k}}{(n+k)!}-\sum_{a=1}^{k}\frac{x^{n+a}}{(n+a)!(k-a)!}\right)$$
• The problem originates from operators. If we let $t^kx^n=D_x^k[x^n]$, then a formal power series can be an operator and for function like $\frac{e^t-1}{t}, \frac{e^t-1-t}{t^2},$ etc, the results are given above in terms of integrals. I was hoping to end up with simpler integral expressions instead of the discrete summations above, but I now have some information i need for other work. Thank you. Commented Nov 3, 2017 at 0:21