`int_(-pi)^pi x^3 cos^3 x dx` is:

Options

A. 0
B. `pi`
C. `pi/4`
D. `2pi`

Answer

To solve the integral \( \int_{-\pi}^{\pi} x^3 \cos^3 x \, dx \), we will use the properties of odd and even functions.

### Step-by-step Solution:

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

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 \( f(-x) \)**:
\[ f(-x) = (-x)^3 \cos^3(-x) \]
- Since \( \cos(-x) = \cos(x) \), we have:
\[ f(-x) = -x^3 \cos^3 x \]
- This simplifies to:
\[ f(-x) = -f(x) \]

4. **Conclusion about the function**: Since \( f(-x) = -f(x) \), the function \( f(x) = x^3 \cos^3 x \) is an odd function.

5. **Use the property of definite integrals**: The property states that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd} \]

6. **Apply the property**:
\[ \int_{-\pi}^{\pi} x^3 \cos^3 x \, dx = 0 \]

### Final Answer:
\[ \int_{-\pi}^{\pi} x^3 \cos^3 x \, dx = 0 \]