Evaluate the following definite integral: `int_(-sqrt(2))^(sqrt(2))(2x^7+3x^6-10 x^5-7x^3-12 x^2+x+1)/(x^2+2)dx`

Answer

To evaluate the definite integral

\[ I = \int_{-\sqrt{2}}^{\sqrt{2}} \frac{2x^7 + 3x^6 - 10x^5 - 7x^3 - 12x^2 + x + 1}{x^2 + 2} \, dx, \]

we can start by analyzing the integrand.

### Step 1: Identify Odd and Even Functions

The integrand can be split into two parts: the numerator and the denominator.

1. **Odd Functions**:
- The terms \(2x^7\), \(-10x^5\), \(-7x^3\), and \(x\) are odd functions.
- The entire numerator can be expressed as a sum of odd and even functions. The odd part will integrate to zero over the symmetric interval \([-a, a]\).

2. **Even Functions**:
- The terms \(3x^6\), \(-12x^2\), and \(1\) are even functions.

### Step 2: Simplify the Integral

Since the odd parts of the numerator will contribute zero to the integral, we can focus on the even part:

\[ I = \int_{-\sqrt{2}}^{\sqrt{2}} \frac{3x^6 - 12x^2 + 1}{x^2 + 2} \, dx. \]

### Step 3: Use the Property of Even Functions

For even functions, we can use the property:

\[ \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx. \]

Thus, we have:

\[ I = 2 \int_{0}^{\sqrt{2}} \frac{3x^6 - 12x^2 + 1}{x^2 + 2} \, dx. \]

### Step 4: Simplify the Integrand

Now we can simplify the integrand:

\[ \frac{3x^6 - 12x^2 + 1}{x^2 + 2} = \frac{3x^6}{x^2 + 2} - \frac{12x^2}{x^2 + 2} + \frac{1}{x^2 + 2}. \]

### Step 5: Break Down the Integral

We can break down the integral into three parts:

\[ I = 2 \left( \int_{0}^{\sqrt{2}} \frac{3x^6}{x^2 + 2} \, dx - 12 \int_{0}^{\sqrt{2}} \frac{x^2}{x^2 + 2} \, dx + \int_{0}^{\sqrt{2}} \frac{1}{x^2 + 2} \, dx \right). \]

### Step 6: Evaluate Each Integral

1. **Integral of \( \frac{3x^6}{x^2 + 2} \)**:
- Use polynomial long division or substitution to evaluate.

2. **Integral of \( \frac{x^2}{x^2 + 2} \)**:
- This can be simplified to \(1 - \frac{2}{x^2 + 2}\).

3. **Integral of \( \frac{1}{x^2 + 2} \)**:
- This can be evaluated using the formula for the integral of \( \frac{1}{x^2 + a^2} \).

### Step 7: Combine Results

After evaluating each integral, combine the results to find \(I\).

### Final Result

The final answer for the definite integral is:

\[ I = \frac{5}{2\sqrt{2}} - \frac{16\sqrt{2}}{5}. \]