`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 use a property of definite integrals. Here is the step-by-step solution:

### Step 1: Use the property of definite integrals
We know that for a function \( f(x) \), the integral from \( a \) to \( b \) can be expressed as:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
In our case, we have \( a = 0 \) and \( b = \pi \). Hence, we can write:
\[ I = \int_{0}^{\pi} [\cot x] \, dx = \int_{0}^{\pi} [\cot(\pi - x)] \, dx \]

### Step 2: Simplify \(\cot(\pi - x)\)
Using the identity \(\cot(\pi - x) = -\cot x\), we can rewrite the integral:
\[ I = \int_{0}^{\pi} [-\cot x] \, dx = -\int_{0}^{\pi} [\cot x] \, dx \]
This gives us:
\[ I = -I \]

### Step 3: Solve for \( I \)
From the equation \( I = -I \), we can add \( I \) to both sides:
\[ 2I = 0 \implies I = 0 \]

### Conclusion
Thus, the value of the integral is:
\[ \int_{0}^{\pi} [\cot x] \, dx = 0 \]