What is `int_(-a)^(a) (x^(3) + sin x) dx` equal to

Options

A. a
B. `2a`
C. `0`
D. `1`

Explanation

To solve the integral \( \int_{-a}^{a} (x^3 + \sin x) \, dx \), we will first determine if the integrand is an even or odd function.

### Step-by-Step Solution:

1. **Identify the function**: Let \( f(x) = x^3 + \sin x \).

2. **Check if the function is odd**: We need to find \( f(-x) \):
\[ f(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin x \]
This can be rewritten as:
\[ f(-x) = - (x^3 + \sin x) = -f(x) \]
Since \( f(-x) = -f(x) \), the function \( f(x) \) is an odd function.

3. **Use the property of integrals of odd functions**: The integral of an odd function over a symmetric interval around zero is zero. Therefore:
\[ \int_{-a}^{a} f(x) \, dx = 0 \]

4. **Conclusion**: Thus, we have:
\[ \int_{-a}^{a} (x^3 + \sin x) \, dx = 0 \]

### Final Answer:
\[ \int_{-a}^{a} (x^3 + \sin x) \, dx = 0 \]