`int_(0)^(pi)[cotx]dx,` where [.] denotes the greatest integer function, is equal to

Options

A. `(pi)/2`
B. `1`
C. `-1`
D. `-(pi)/2`

Explanation

To solve the integral \( I = \int_{0}^{\pi} [\cot x] \, dx \), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Define the Integral
Let
\[ I = \int_{0}^{\pi} [\cot x] \, dx \]

### Step 2: Use the Property of Definite Integrals
We can use the property of definite integrals:
\[ \int_{0}^{A} f(x) \, dx = \int_{0}^{A} f(A - x) \, dx \]
In our case, we have:
\[ I = \int_{0}^{\pi} [\cot(\pi - x)] \, dx \]

### Step 3: Simplify \(\cot(\pi - x)\)
Using the identity \(\cot(\pi - x) = -\cot x\), we rewrite the integral:
\[ I = \int_{0}^{\pi} [ -\cot x ] \, dx \]

### Step 4: Combine the Integrals
Now we can add the two expressions for \(I\):
\[ 2I = \int_{0}^{\pi} [\cot x] \, dx + \int_{0}^{\pi} [-\cot x] \, dx \]
This simplifies to:
\[ 2I = \int_{0}^{\pi} \left( [\cot x] + [-\cot x] \right) \, dx \]

### Step 5: Apply the Property of the Greatest Integer Function
From the property of the greatest integer function, we know:
\[ [\cot x] + [-\cot x] = -1 \quad \text{(for } \cot x \text{ not an integer)} \]
Thus, we have:
\[ 2I = \int_{0}^{\pi} -1 \, dx \]

### Step 6: Evaluate the Integral
Now we can evaluate the integral:
\[ 2I = -\int_{0}^{\pi} 1 \, dx = -[x]_{0}^{\pi} = -(\pi - 0) = -\pi \]

### Step 7: Solve for \(I\)
Now, we can solve for \(I\):
\[ I = \frac{-\pi}{2} \]

### Final Answer
Thus, the value of the integral is:
\[ I = -\frac{\pi}{2} \]