The value of the integral `int_(-10)^(0) (|(2[x])/(3x-[x])|)/(((2[x])/(3x-[x])))dx` where [.] denotes GIF

Options

A. `(28)/(3)`
B. 0
C. 10
D. -10

Answer

To solve the integral

\[ I = \int_{-10}^{0} \frac{|\frac{2[x]}{3x - [x]}|}{\frac{2[x]}{3x - [x]}} \, dx, \]

where \([x]\) denotes the greatest integer function (GIF), we will break down the integral into manageable parts.

### Step 1: Break down the integral

We can break the integral into intervals where the behavior of the greatest integer function \([x]\) is constant. The intervals will be:

- From \(-10\) to \(-9\)
- From \(-9\) to \(-8\)
- From \(-8\) to \(-7\)
- From \(-7\) to \(-6\)
- From \(-6\) to \(-5\)
- From \(-5\) to \(-4\)
- From \(-4\) to \(-3\)
- From \(-3\) to \(-2\)
- From \(-2\) to \(-1\)
- From \(-1\) to \(0\)

Thus, we can express the integral as:

\[ I = \sum_{n=-10}^{-1} \int_{n}^{n+1} \frac{|\frac{2n}{3x - n}|}{\frac{2n}{3x - n}} \, dx. \]

### Step 2: Evaluate each integral

For each interval \([-n, -n+1]\), \([x] = n\). Therefore, we can simplify the expression:

\[ \frac{|\frac{2n}{3x - n}|}{\frac{2n}{3x - n}} = \text{sgn}\left(\frac{2n}{3x - n}\right). \]

This means we need to analyze the sign of \(\frac{2n}{3x - n}\).

### Step 3: Determine the sign

1. **For \(x \in [-10, -9]\)**: Here, \(n = -10\).
- \(\frac{2(-10)}{3x - (-10)} = \frac{-20}{3x + 10}\).
- For \(x = -10\), this is undefined. As \(x\) approaches \(-10\) from the right, the expression is negative.
- Thus, the integral evaluates to \(-1\) over this interval.

2. **For \(x \in [-9, -8]\)**: Here, \(n = -9\).
- \(\frac{2(-9)}{3x + 9}\).
- Similar analysis shows this is also negative, yielding \(-1\).

Continuing this analysis through all intervals, we find that:

- For each interval from \([-10, -9]\) to \([-1, 0]\), the integral evaluates to \(-1\).

### Step 4: Sum the contributions

Since there are 10 intervals from \(-10\) to \(0\):

\[ I = \sum_{n=-10}^{-1} (-1) = 10 \times (-1) = -10. \]

### Conclusion

The value of the integral is:

\[ \boxed{-10}. \]