Evaluate: `int_(-2)^2 (x^3+x)dx`

Answer

To evaluate the integral \( \int_{-2}^{2} (x^3 + x) \, dx \), we can follow these steps:

### Step 1: Identify the function
The function we are integrating is \( f(x) = x^3 + x \).

### Step 2: Check if the function is odd or even
To determine if the function is odd or even, we need to evaluate \( f(-x) \):
\[ f(-x) = (-x)^3 + (-x) = -x^3 - x = -(x^3 + x) = -f(x) \]
Since \( f(-x) = -f(x) \), the function \( f(x) \) is an odd function.

### Step 3: Use the property of definite integrals
A property of definite integrals states that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd} \]
In our case, since \( f(x) = x^3 + x \) is odd and we are integrating from \(-2\) to \(2\), we can conclude:
\[ \int_{-2}^{2} (x^3 + x) \, dx = 0 \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{-2}^{2} (x^3 + x) \, dx = 0 \]

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