The number of integral values in the range of the function `f(x)=sin^(-1)x-cot^(-1)x+x^(2)+2x +6` is

Options

A. 10
B. 11
C. 12
D. 8

Explanation

To find the number of integral values in the range of the function \( f(x) = \sin^{-1}(x) - \cot^{-1}(x) + x^2 + 2x + 6 \), we will follow these steps:

### Step 1: Rewrite the function
We know that \( \cot^{-1}(x) + \tan^{-1}(x) = \frac{\pi}{2} \). Therefore, we can rewrite \( f(x) \) as:
\[ f(x) = \sin^{-1}(x) - \left(\frac{\pi}{2} - \tan^{-1}(x)\right) + x^2 + 2x + 6 \]
This simplifies to:
\[ f(x) = \sin^{-1}(x) + \tan^{-1}(x) + x^2 + 2x + 6 - \frac{\pi}{2} \]

### Step 2: Determine the domain of the function
The domain of \( \sin^{-1}(x) \) is \( x \in [-1, 1] \). Therefore, the domain of \( f(x) \) is also \( x \in [-1, 1] \).

### Step 3: Find the minimum and maximum values of \( f(x) \)
To find the minimum and maximum values of \( f(x) \) within the domain, we will evaluate \( f(x) \) at the endpoints of the interval:

1. **Evaluate at \( x = -1 \)**:
\[ f(-1) = \sin^{-1}(-1) + \tan^{-1}(-1) + (-1)^2 + 2(-1) + 6 - \frac{\pi}{2} \]
\[ = -\frac{\pi}{2} - \frac{\pi}{4} + 1 - 2 + 6 - \frac{\pi}{2} \]
\[ = 5 - \frac{5\pi}{4} \]

2. **Evaluate at \( x = 1 \)**:
\[ f(1) = \sin^{-1}(1) + \tan^{-1}(1) + (1)^2 + 2(1) + 6 - \frac{\pi}{2} \]
\[ = \frac{\pi}{2} + \frac{\pi}{4} + 1 + 2 + 6 - \frac{\pi}{2} \]
\[ = 9 + \frac{\pi}{4} \]

### Step 4: Determine the range of \( f(x) \)
The minimum value of \( f(x) \) occurs at \( x = -1 \):
\[ f(-1) = 5 - \frac{5\pi}{4} \]
The maximum value of \( f(x) \) occurs at \( x = 1 \):
\[ f(1) = 9 + \frac{\pi}{4} \]
Thus, the range of \( f(x) \) is:
\[ \left[ 5 - \frac{5\pi}{4}, 9 + \frac{\pi}{4} \right] \]

### Step 5: Calculate the integral values in the range
To find the number of integral values in the range:
1. Calculate \( 5 - \frac{5\pi}{4} \):
- Approximating \( \pi \approx 3.14 \):
\[ 5 - \frac{5 \times 3.14}{4} \approx 5 - 3.925 = 1.075 \]

2. Calculate \( 9 + \frac{\pi}{4} \):
- Approximating \( \frac{\pi}{4} \approx 0.785 \):
\[ 9 + 0.785 \approx 9.785 \]

The integral values in the range from approximately \( 1.075 \) to \( 9.785 \) are \( 2, 3, 4, 5, 6, 7, 8, 9 \).

### Conclusion
The total number of integral values in the range of \( f(x) \) is \( 8 \).