`int_(-pi//2)^(pi//2)sin^(5)xcos^(4)xdx`

Answer

To solve the integral \( I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^5 x \cos^4 x \, dx \), we will determine if the integrand is an odd function. If it is, the integral over a symmetric interval around zero will equal zero.

### Step-by-Step Solution:

1. **Define the Function:**
Let \( f(x) = \sin^5 x \cos^4 x \).

2. **Check for Odd Function:**
We need to check if \( f(-x) = -f(x) \).
\[ f(-x) = \sin^5(-x) \cos^4(-x) \]
Using the properties of sine and cosine:
\[ \sin(-x) = -\sin(x) \quad \text{and} \quad \cos(-x) = \cos(x) \]
Therefore,
\[ f(-x) = (-\sin x)^5 (\cos x)^4 = -\sin^5 x \cos^4 x = -f(x) \]

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

4. **Evaluate the Integral:**
By the property of definite integrals, if \( f(x) \) is odd, then:
\[ \int_{-a}^{a} f(x) \, dx = 0 \]
Applying this to our integral:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^5 x \cos^4 x \, dx = 0 \]

### Final Answer:
\[ I = 0 \]