Step-by-step Solution:

1. Define the Greatest Integer Function: The function \([x]\) is defined as: \[ [x] = n \quad \text{for} \quad x \in [n, n+1), \quad n \in \mathbb{Z} \] In the interval \([0, 9)\), the function \([x]\) takes integer values from 0 to 8. 2. Break the Integral into Subintervals: The integral can be expressed as the sum of integrals over each subinterval where \([x]\) is constant: \[ \int_{0}^{9} [x] \, dx = \int_{0}^{1} 0 \, dx + \int_{1}^{2} 1 \, dx + \int_{2}^{3} 2 \, dx + \cdots + \int_{8}^{9} 8 \, dx \] 3. Evaluate Each Integral: Each integral over a unit interval is straightforward: \[ \int_{n}^{n+1} n \, dx = n \cdot (n+1 - n) = n \cdot 1 = n \] Therefore: \[ \int_{0}^{9} [x] \, dx = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 \] 4. Calculate the Sum: The sum of the first 8 natural numbers is: \[ 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = \frac{8(8+1)}{2} = 36 \] Therefore, the value of the integral is: \[ \boxed{36} \]