Absolute value of sum of all integers in the domain of `f(x)=cot^(-1)sqrt((x+3)x)+cos^(-1)sqrt(x^2+3x+1)` is_______

Explanation

To find the absolute value of the sum of all integers in the domain of the function \( f(x) = \cot^{-1}(\sqrt{(x+3)x}) + \cos^{-1}(\sqrt{x^2 + 3x + 1}) \), we need to determine the domain of the function first.

### Step 1: Determine the domain of \( \cot^{-1}(\sqrt{(x+3)x}) \)

The expression inside the cotangent inverse function, \( \sqrt{(x+3)x} \), must be non-negative. Therefore, we need:

\[ (x+3)x \geq 0 \]

This inequality holds when either both factors are non-negative or both are non-positive.

1. **Both factors non-negative:**
- \( x + 3 \geq 0 \) implies \( x \geq -3 \)
- \( x \geq 0 \)

The intersection gives us \( x \geq 0 \).

2. **Both factors non-positive:**
- \( x + 3 \leq 0 \) implies \( x \leq -3 \)
- \( x \leq 0 \)

The intersection gives us \( x \leq -3 \).

Thus, the solution to \( (x+3)x \geq 0 \) is:

\[ x \in (-\infty, -3] \cup [0, \infty) \]

### Step 2: Determine the domain of \( \cos^{-1}(\sqrt{x^2 + 3x + 1}) \)

The expression inside the cosine inverse function, \( \sqrt{x^2 + 3x + 1} \), must lie between 0 and 1, inclusive:

\[ 0 \leq \sqrt{x^2 + 3x + 1} \leq 1 \]

Squaring the inequality gives:

\[ 0 \leq x^2 + 3x + 1 \leq 1 \]

1. **For \( x^2 + 3x + 1 \geq 0 \):**
- The roots of \( x^2 + 3x + 1 = 0 \) can be found using the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-3 \pm \sqrt{5}}{2} \]
The quadratic opens upwards, so it is non-negative outside the interval formed by its roots.

2. **For \( x^2 + 3x + 1 \leq 1 \):**
- Rearranging gives:
\[ x^2 + 3x \leq 0 \]
Factoring gives:
\[ x(x + 3) \leq 0 \]
This holds for \( x \in [-3, 0] \).

### Step 3: Combine the domains

From the two parts, we have:

- From \( \cot^{-1}(\sqrt{(x+3)x}) \): \( x \in (-\infty, -3] \cup [0, \infty) \)
- From \( \cos^{-1}(\sqrt{x^2 + 3x + 1}) \): \( x \in [-3, 0] \)

The intersection of these two domains is:

\[ x \in [-3, 0] \]

### Step 4: Find the integers in the domain

The integers in the interval \( [-3, 0] \) are:

\[ -3, -2, -1, 0 \]

### Step 5: Calculate the sum of these integers

Calculating the sum:

\[ -3 + (-2) + (-1) + 0 = -6 \]

### Step 6: Find the absolute value

The absolute value of the sum is:

\[ | -6 | = 6 \]

Thus, the final answer is:

\[ \boxed{6} \]