Sum of a power series

Question

I have to find the sum of this series $$\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}$$

Using integral, I got $$\int\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n!}=\sum_{n=1}^{+\infty}\frac{x^{n}}{n\cdot n!}$$

I know that $$\sum_{n=1}^{+\infty}\frac{x^{n}}{n!}=e^{x}$$

But my problem is about this $n\cdot n!$. How can I solve it?

Thanks!