`int_(-pi//2)^(pi//2)sin^(9)x cos^(5)xdx` equals

Options

A. `(1)/(20)`
B. 20
C. 0
D. `(1)/(330)`

Answer

To solve the integral \( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^9 x \cos^5 x \, dx \), we will use the properties of definite integrals, specifically focusing on whether the integrand is an odd or even function.

### Step-by-Step Solution:

1. **Identify the Integral**:
We need to evaluate the integral:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^9 x \cos^5 x \, dx \]

2. **Check for Odd or Even Function**:
To determine if the function \( f(x) = \sin^9 x \cos^5 x \) is odd or even, we will compute \( f(-x) \):
\[ f(-x) = \sin^9(-x) \cos^5(-x) \]
Using the properties of sine and cosine:
\[ \sin(-x) = -\sin(x) \quad \text{and} \quad \cos(-x) = \cos(x) \]
Therefore,
\[ f(-x) = (-\sin x)^9 \cdot (\cos x)^5 = -\sin^9 x \cos^5 x = -f(x) \]

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

4. **Apply the Property of Definite Integrals**:
The property states that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{for odd } f(x) \]
Thus, we have:
\[ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^9 x \cos^5 x \, dx = 0 \]

### Final Answer:
\[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^9 x \cos^5 x \, dx = 0 \]