`int_(1)^(10pi)([sec^(-1)x]+[cot^(-1)x])dx`, where [.] denotes the greatest integer function, is equal to:

Options

A. `10 pi` - sec1
B. `10pi +` secl
C. `10pi - secl + cotl`
D. secl + cotl

Answer

To solve the integral \( \int_{1}^{10\pi} \left[ \sec^{-1} x + \cot^{-1} x \right] dx \), where \([.]\) denotes the greatest integer function, we will follow these steps:

### Step 1: Understanding the Functions
First, we need to understand the behavior of the functions \( \sec^{-1} x \) and \( \cot^{-1} x \).

- The function \( \sec^{-1} x \) is defined for \( x \geq 1 \) and is increasing. At \( x = 1 \), \( \sec^{-1}(1) = 0 \).
- The function \( \cot^{-1} x \) is defined for \( x > 0 \) and is decreasing. At \( x = 1 \), \( \cot^{-1}(1) = \frac{\pi}{4} \).

### Step 2: Finding the Values at the Bounds
Next, we calculate the values of \( \sec^{-1} x + \cot^{-1} x \) at the bounds of the integral.

At \( x = 1 \):
\[ \sec^{-1}(1) + \cot^{-1}(1) = 0 + \frac{\pi}{4} = \frac{\pi}{4} \]

As \( x \) increases, \( \sec^{-1} x \) increases and \( \cot^{-1} x \) decreases.

### Step 3: Behavior of the Sum
For \( x \) in the interval \( [1, 10\pi] \):
- As \( x \) approaches \( 10\pi \), \( \sec^{-1} x \) approaches a value greater than \( \frac{\pi}{4} \) and \( \cot^{-1} x \) approaches \( 0 \).
- Therefore, \( \sec^{-1} x + \cot^{-1} x \) will vary between \( \frac{\pi}{4} \) and a value greater than \( \frac{\pi}{4} \).

### Step 4: Applying the Greatest Integer Function
Now, we need to find the greatest integer function value of \( \sec^{-1} x + \cot^{-1} x \):
- At \( x = 1 \), \( \sec^{-1}(1) + \cot^{-1}(1) = \frac{\pi}{4} \approx 0.785 \).
- As \( x \) increases, \( \sec^{-1} x + \cot^{-1} x \) will remain less than \( 1 \) for \( x < 2 \) (since \( \sec^{-1} x \) starts from 0 and \( \cot^{-1} x \) starts from \( \frac{\pi}{4} \)).
- Therefore, for \( 1 \leq x < 2 \), the greatest integer function \( \left[ \sec^{-1} x + \cot^{-1} x \right] = 0 \).

### Step 5: Evaluating the Integral
Since \( \left[ \sec^{-1} x + \cot^{-1} x \right] = 0 \) for \( x \) in the interval \( [1, 10\pi] \), the integral simplifies to:
\[ \int_{1}^{10\pi} 0 \, dx = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{0} \]