Evaluate `int _(-1)^(1) (x^(3)+5x+ sin x) d x `

Answer

To evaluate the integral \(\int_{-1}^{1} (x^{3} + 5x + \sin x) \, dx\), we will determine whether the integrand is an even function or an odd function.

### Step-by-Step Solution:

1. **Identify the function**: Let \( f(x) = x^{3} + 5x + \sin x \).

2. **Check if the function is odd or even**:
- A function \( f(x) \) is **even** if \( f(-x) = f(x) \).
- A function \( f(x) \) is **odd** if \( f(-x) = -f(x) \).

3. **Calculate \( f(-x) \)**:
\[ f(-x) = (-x)^{3} + 5(-x) + \sin(-x) \]
Simplifying this:
\[ f(-x) = -x^{3} - 5x - \sin x \]

4. **Compare \( f(-x) \) with \(-f(x)\)**:
- We know \( f(x) = x^{3} + 5x + \sin x \).
- Now, calculate \(-f(x)\):
\[ -f(x) = -(x^{3} + 5x + \sin x) = -x^{3} - 5x - \sin x \]
- We see that \( f(-x) = -f(x) \).

5. **Conclusion about the function**: Since \( f(-x) = -f(x) \), the function \( f(x) \) is an **odd function**.

6. **Evaluate the integral**:
- The integral of an odd function over a symmetric interval \([-a, a]\) is zero:
\[ \int_{-1}^{1} (x^{3} + 5x + \sin x) \, dx = 0 \]

### Final Answer:
\[ \int_{-1}^{1} (x^{3} + 5x + \sin x) \, dx = 0 \]