`int _(-5) ^(5) (x )/(x ^(2) + 7) dx = 10`. True or False.

Answer

To determine whether the integral

\[ \int_{-5}^{5} \frac{x}{x^2 + 7} \, dx = 10 \]

is true or false, we can analyze the integral step by step.

### Step 1: Identify the Function
The integrand is

\[ f(x) = \frac{x}{x^2 + 7}. \]

### Step 2: Check for Odd Function
To check if \( f(x) \) is an odd function, we need to evaluate \( f(-x) \):

\[ f(-x) = \frac{-x}{(-x)^2 + 7} = \frac{-x}{x^2 + 7} = -f(x). \]

Since \( f(-x) = -f(x) \), this confirms that \( f(x) \) is indeed an odd function.

### Step 3: Use the Property of Odd Functions
The property of odd functions states that the integral of an odd function over a symmetric interval around zero is zero:

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

In our case, since we are integrating from \(-5\) to \(5\), we can conclude:

\[ \int_{-5}^{5} \frac{x}{x^2 + 7} \, dx = 0. \]

### Step 4: Compare with Given Value
The question states that the integral equals \(10\):

\[ \int_{-5}^{5} \frac{x}{x^2 + 7} \, dx = 10. \]

Since we found that the integral actually equals \(0\), we conclude that the statement is false.

### Final Answer
Thus, the statement is **False**.

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