The value of `int_(-pi//2)^(pi//2)[ cot^(-1)x] dx` (where ,[.] denotes greatest integer function) is equal to

Options

A. (a)`pi+ cot 1`
B. (b)`pi+ cot 2`
C. (c)`pi+ cot1 + cot2`
D. (d)`cot 1+cot 2`

Answer

To solve the integral \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cot^{-1}(x) \, dx \), we can use the properties of the cotangent inverse function and symmetry.

### Step 1: Define the Integral
Let \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cot^{-1}(x) \, dx \).

### Step 2: Use the Property of Cotangent Inverse
We know that:
\[ \cot^{-1}(-x) = \pi - \cot^{-1}(x) \quad \text{for } x \in \mathbb{R} \]
Thus, we can express the integral as:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cot^{-1}(-x) \, dx \]
Substituting the property we have:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \pi - \cot^{-1}(x) \right) \, dx \]

### Step 3: Split the Integral
Now, we can split the integral into two parts:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi \, dx - \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cot^{-1}(x) \, dx \]
The first integral evaluates to:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \pi \, dx = \pi \left( \frac{\pi}{2} - \left(-\frac{\pi}{2}\right) \right) = \pi \cdot \pi = \pi^2 \]
Thus, we have:
\[ I = \pi^2 - I \]

### Step 4: Solve for \( I \)
Now, we can solve for \( I \):
\[ 2I = \pi^2 \]
\[ I = \frac{\pi^2}{2} \]

### Step 5: Apply the Greatest Integer Function
Since the problem states that we need to apply the greatest integer function, we need to find \( \lfloor I \rfloor \):
\[ \lfloor I \rfloor = \lfloor \frac{\pi^2}{2} \rfloor \]
Using the approximation \( \pi \approx 3.14 \):
\[ \frac{\pi^2}{2} \approx \frac{(3.14)^2}{2} \approx \frac{9.8596}{2} \approx 4.9298 \]
Thus, \( \lfloor \frac{\pi^2}{2} \rfloor = 4 \).

### Final Answer
The value of \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cot^{-1}(x) \, dx \) (where the greatest integer function is applied) is:
\[ \lfloor I \rfloor = 4 \]