`int_- 10^0 (|(2[x])/(3x-[x]|)/(2[x])/(3x-[x]))dx` is equal to (where [*] denotes greatest integer function.) is equal to (where [*] denotes greatest integer function.)

Options

A. `28/3`
B. `1/3`
C. `0`
D. None of these

Answer

To solve the integral

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

where \([x]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understanding the Greatest Integer Function
For \(x < 0\), the greatest integer function \([x]\) is equal to \(-1\) for \(-1 < x < 0\), \(-2\) for \(-2 < x < -1\), \(-3\) for \(-3 < x < -2\), and so on.

### Step 2: Splitting the Integral
We will split the integral into segments based on the intervals defined by the greatest integer function:
1. From \(-10\) to \(-3\)
2. From \(-3\) to \(-2\)
3. From \(-2\) to \(-1\)
4. From \(-1\) to \(0\)

### Step 3: Evaluating Each Segment
1. **For \(x \in [-10, -3)\)**: Here, \([x] = -4\).
\[ \int_{-10}^{-3} \frac{|2[-4]|}{3x - [-4]} \, dx = \int_{-10}^{-3} \frac{8}{3x + 4} \, dx \]

2. **For \(x \in [-3, -2)\)**: Here, \([x] = -3\).
\[ \int_{-3}^{-2} \frac{|2[-3]|}{3x - [-3]} \, dx = \int_{-3}^{-2} \frac{6}{3x + 3} \, dx \]

3. **For \(x \in [-2, -1)\)**: Here, \([x] = -2\).
\[ \int_{-2}^{-1} \frac{|2[-2]|}{3x - [-2]} \, dx = \int_{-2}^{-1} \frac{4}{3x + 2} \, dx \]

4. **For \(x \in [-1, 0)\)**: Here, \([x] = -1\).
\[ \int_{-1}^{0} \frac{|2[-1]|}{3x - [-1]} \, dx = \int_{-1}^{0} \frac{2}{3x + 1} \, dx \]

### Step 4: Calculating Each Integral
1. **Integral from \(-10\) to \(-3\)**:
\[ \int_{-10}^{-3} \frac{8}{3x + 4} \, dx = \left[ \frac{8}{3} \ln |3x + 4| \right]_{-10}^{-3} \]

2. **Integral from \(-3\) to \(-2\)**:
\[ \int_{-3}^{-2} \frac{6}{3x + 3} \, dx = \left[ 2 \ln |3x + 3| \right]_{-3}^{-2} \]

3. **Integral from \(-2\) to \(-1\)**:
\[ \int_{-2}^{-1} \frac{4}{3x + 2} \, dx = \left[ \frac{4}{3} \ln |3x + 2| \right]_{-2}^{-1} \]

4. **Integral from \(-1\) to \(0\)**:
\[ \int_{-1}^{0} \frac{2}{3x + 1} \, dx = \left[ \frac{2}{3} \ln |3x + 1| \right]_{-1}^{0} \]

### Step 5: Summing the Results
After calculating each integral, we sum them to find the total value of the original integral.

### Final Answer
The final result of the integral is:

\[ \int_{-10}^{0} \frac{|2[x]|}{3x - [x]} \, dx = \frac{28}{3}. \]