If `f(x)=int(x^8+4)/(x^4-2x^2+2)dxa n df(0)=0,t h e n` `f(x)` is an odd function `f(x)` has range `R` `f(x)` has at least one real root `f(x)` is a monotonic function.

Options

A. `f(x)` is an odd function
B. `f(x)` has range R
C. `f(x)` has at least one real root
D. `f(x)` is a monotonic function

Explanation

To solve the problem, we need to analyze the function given by the integral:

\[ f(x) = \int \frac{x^8 + 4}{x^4 - 2x^2 + 2} \, dx \]

### Step 1: Simplify the integrand

First, we can simplify the expression in the integrand:

\[ f(x) = \int \frac{x^8 + 4}{x^4 - 2x^2 + 2} \, dx \]

We can rewrite \(x^8 + 4\) as \(x^8 + 4x^4 + 4 - 4x^4\):

\[ f(x) = \int \frac{(x^4 + 2)^2 - (2x^2)^2}{x^4 - 2x^2 + 2} \, dx \]

### Step 2: Factor the numerator

Now, we can factor the numerator using the difference of squares:

\[ f(x) = \int \left( \frac{(x^4 + 2 + 2x^2)(x^4 + 2 - 2x^2)}{x^4 - 2x^2 + 2} \right) \, dx \]

### Step 3: Separate the integral

We can separate the integral into two parts:

\[ f(x) = \int (x^4 + 2 + 2x^2) \, dx - \int (x^4 + 2 - 2x^2) \, dx \]

### Step 4: Calculate the integrals

Now we can compute the integrals:

1. For \( \int (x^4 + 2 + 2x^2) \, dx \):

\[ = \frac{x^5}{5} + 2x + \frac{2x^3}{3} + C_1 \]

2. For \( \int (x^4 + 2 - 2x^2) \, dx \):

\[ = \frac{x^5}{5} + 2x - \frac{2x^3}{3} + C_2 \]

### Step 5: Combine the results

Combining both results, we find:

\[ f(x) = \left( \frac{x^5}{5} + 2x + \frac{2x^3}{3} \right) - \left( \frac{x^5}{5} + 2x - \frac{2x^3}{3} \right) + C \]

This simplifies to:

\[ f(x) = \frac{4x^3}{3} + C \]

### Step 6: Apply the condition \(f(0) = 0\)

Given that \(f(0) = 0\):

\[ f(0) = \frac{4(0)^3}{3} + C = 0 \implies C = 0 \]

Thus, we have:

\[ f(x) = \frac{4x^3}{3} \]

### Step 7: Analyze the function

1. **Odd Function**: Since \(f(-x) = -f(x)\), \(f(x)\) is an odd function.
2. **Range**: The function \(f(x) = \frac{4x^3}{3}\) has a range of all real numbers \(R\).
3. **Real Roots**: The equation \(f(x) = 0\) has at least one real root at \(x = 0\).
4. **Monotonic Function**: The derivative \(f'(x) = 4x^2\) is non-negative for all \(x\), indicating that \(f(x)\) is a monotonic function.

### Conclusion

Thus, all four statements about \(f(x)\) are true:

- \(f(x)\) is an odd function.
- \(f(x)\) has range \(R\).
- \(f(x)\) has at least one real root.
- \(f(x)\) is a monotonic function.