`int_(-1)^(1)sin^(11)xdx` is equal to

Options

A. `10/11`
B. `2`
C. 1
D. 0

Answer

To solve the integral \( I = \int_{-1}^{1} \sin^{11}(x) \, dx \), we will use the property of definite integrals which states that for an odd function \( f(x) \), the integral from \(-a\) to \(a\) is zero.

### Step-by-step Solution:

1. **Identify the Function**:
We have \( f(x) = \sin^{11}(x) \).

2. **Check if the Function is Odd**:
A function \( f(x) \) is odd if \( f(-x) = -f(x) \).
Let's compute \( f(-x) \):
\[ f(-x) = \sin^{11}(-x) = (-\sin(x))^{11} = -\sin^{11}(x) \]
Since \( f(-x) = -f(x) \), the function \( \sin^{11}(x) \) is indeed an odd function.

3. **Apply the Property of Definite Integrals**:
For any odd function \( f(x) \):
\[ \int_{-a}^{a} f(x) \, dx = 0 \]
Here, \( a = 1 \), so we can write:
\[ I = \int_{-1}^{1} \sin^{11}(x) \, dx = 0 \]

4. **Conclusion**:
The value of the integral \( I \) is:
\[ I = 0 \]

### Final Answer:
\[ \int_{-1}^{1} \sin^{11}(x) \, dx = 0 \]