Example 4 Find the integer solution of the equation $\left[\frac{x}{1!}\right]+\left[\frac{x}{2!}\right]+\left[\frac{x}{3!}\right]+\cdots+$ $\left[\frac{x}{10!}\right]=1001$.
Solution: $\because$ the solution $x$ of the equation is a natural number, and $xx+\frac{x}{2}+\frac{x}{6}+\frac{x}{24}+\frac{x}{120}-4 \\
=\frac{103}{60} x-4, \therefore 583<x<585, x=584,
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Verification shows that $x=584$ is the unique solution of the equation.