`int_(-1)^1 sin^5 x cos^4 x dx` is :

Options

A. 1
B. -1
C. 0
D. None of these

Answer

To solve the integral \( I = \int_{-1}^{1} \sin^5 x \cos^4 x \, dx \), we can utilize the property of definite integrals that states:

\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(-x) = -f(x) \]

This means that if the function \( f(x) \) is odd, the integral over a symmetric interval around zero will evaluate to zero.

### Step-by-Step Solution:

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

2. **Check for Oddness**: We need to check whether \( f(-x) = -f(x) \):
- Calculate \( f(-x) \):
\[ f(-x) = \sin^5(-x) \cos^4(-x) \]
- Using the properties of sine and cosine:
\[ \sin(-x) = -\sin(x) \quad \text{and} \quad \cos(-x) = \cos(x) \]
- Substitute these into \( f(-x) \):
\[ f(-x) = (-\sin x)^5 (\cos x)^4 = -\sin^5 x \cos^4 x = -f(x) \]

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

4. **Evaluate the Integral**: Now, applying the property of definite integrals:
\[ I = \int_{-1}^{1} f(x) \, dx = 0 \]

Thus, the value of the integral is:
\[ \boxed{0} \]