The value of the integral `int_(0)^(0.9) [x-2[x]] dx`, where [.] denotes the greatest integer function, is

Options

A. 0.9
B. 0
C. 1.8
D. -0.9

Explanation

To solve the integral \( I = \int_{0}^{0.9} \left[ x - 2 \left[ x \right] \right] dx \), where \(\left[ x \right]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Understand the greatest integer function
The greatest integer function \(\left[ x \right]\) gives the largest integer less than or equal to \(x\). For \(x\) in the range \([0, 0.9]\), we have:
\[ \left[ x \right] = 0 \]
because \(0 \leq x < 1\).

### Step 2: Substitute the greatest integer function into the integral
Since \(\left[ x \right] = 0\) for all \(x\) in \([0, 0.9]\), we can substitute this into the integral:
\[ I = \int_{0}^{0.9} \left[ x - 2 \cdot 0 \right] dx = \int_{0}^{0.9} x \, dx \]

### Step 3: Evaluate the integral
Now we need to evaluate the integral:
\[ I = \int_{0}^{0.9} x \, dx \]
The integral of \(x\) is:
\[ \int x \, dx = \frac{x^2}{2} \]
Now we evaluate this from \(0\) to \(0.9\):
\[ I = \left[ \frac{x^2}{2} \right]_{0}^{0.9} = \frac{(0.9)^2}{2} - \frac{(0)^2}{2} = \frac{0.81}{2} = 0.405 \]

### Conclusion
Thus, the value of the integral is:
\[ I = 0.405 \]