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 start with the function:
\[
f(x) = \sin^{-1}(x) - \cot^{-1}(x) + x^2 + 2x + 6
\]
We can rewrite \( \cot^{-1}(x) \) as:
\[
\cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x)
\]
Thus, we can express \( f(x) \) as:
\[
f(x) = \sin^{-1}(x) + \tan^{-1}(x) - \frac{\pi}{2} + x^2 + 2x + 6
\]
### Step 2: Simplify the quadratic part
Next, we can simplify the quadratic expression \( x^2 + 2x + 6 \):
\[
x^2 + 2x + 6 = (x + 1)^2 + 5
\]
So, we can rewrite \( f(x) \) as:
\[
f(x) = \sin^{-1}(x) + \tan^{-1}(x) - \frac{\pi}{2} + (x + 1)^2 + 5
\]
### Step 3: Determine the domain
The domain of \( \sin^{-1}(x) \) is \( [-1, 1] \). Therefore, the domain of \( f(x) \) is also \( [-1, 1] \).
### Step 4: Analyze the behavior of the function
Since \( \sin^{-1}(x) \) and \( \tan^{-1}(x) \) are both increasing functions on the interval \( [-1, 1] \), and \( (x + 1)^2 + 5 \) is also increasing, we conclude that \( f(x) \) is an increasing function on the interval \( [-1, 1] \).
### Step 5: Calculate the endpoints of the function
Now we will calculate \( f(-1) \) and \( f(1) \):
1. **Calculate \( f(-1) \)**:
\[
f(-1) = \sin^{-1}(-1) - \cot^{-1}(-1) + (-1)^2 + 2(-1) + 6
\]
\[
= -\frac{\pi}{2} - \left(\frac{\pi}{2} - \tan^{-1}(-1)\right) + 1 - 2 + 6
\]
\[
= -\frac{\pi}{2} - \frac{\pi}{2} + \frac{\pi}{4} + 5
\]
\[
= -\pi + \frac{\pi}{4} + 5 = -\frac{4\pi}{4} + \frac{\pi}{4} + 5 = -\frac{3\pi}{4} + 5
\]
2. **Calculate \( f(1) \)**:
\[
f(1) = \sin^{-1}(1) - \cot^{-1}(1) + (1)^2 + 2(1) + 6
\]
\[
= \frac{\pi}{2} - \frac{\pi}{4} + 1 + 2 + 6
\]
\[
= \frac{\pi}{2} - \frac{\pi}{4} + 9 = \frac{2\pi}{4} - \frac{\pi}{4} + 9 = \frac{\pi}{4} + 9
\]
### Step 6: Find the range of \( f(x) \)
Now we have:
\[
f(-1) = -\frac{3\pi}{4} + 5 \quad \text{and} \quad f(1) = \frac{\pi}{4} + 9
\]
To find the range of \( f(x) \):
- Lower bound: \( -\frac{3\pi}{4} + 5 \)
- Upper bound: \( \frac{\pi}{4} + 9 \)
### Step 7: Calculate the range numerically
Using approximate values for \( \pi \approx 3.14 \):
- Lower bound: \( -\frac{3 \times 3.14}{4} + 5 \approx -2.355 + 5 \approx 2.645 \)
- Upper bound: \( \frac{3.14}{4} + 9 \approx 0.785 + 9 \approx 9.785 \)
### Step 8: Count the integral values
The range of \( f(x) \) is approximately \( [2.645, 9.785] \). The integral values in this range are \( 3, 4, 5, 6, 7, 8, 9 \).
Thus, the number of integral values is:
\[
\text{Count} = 9 - 3 + 1 = 7
\]
### Final Answer
The number of integral values in the range of the function \( f(x) \) is \( 7 \).
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