The value of the integral `int_(-1)^(1) sin^(11)x" dx"` is

Options

A. `(10)/(11).(8)/(9).(6)/(7).(4)/(5).(2)/(3)`
B. `(10)/(11).(8)/(9).(6)/(7).(4)/(5).(2)/(3).(pi)/(2)`
C. 1
D. 0

Answer

To solve the integral \( I = \int_{-1}^{1} \sin^{11}(x) \, dx \), we can follow these steps:

### Step 1: Identify the function
Let \( f(x) = \sin^{11}(x) \).

### Step 2: Check if the function is even or odd
To determine if \( f(x) \) is even or odd, we need to compute \( f(-x) \):
\[ f(-x) = \sin^{11}(-x) = (-\sin(x))^{11} = -\sin^{11}(x) = -f(x) \]
Since \( f(-x) = -f(x) \), the function \( f(x) \) is an odd function.

### Step 3: Apply the property of definite integrals
We know that the integral of an odd function over a symmetric interval around zero is zero:
\[ \int_{-a}^{a} f(x) \, dx = 0 \quad \text{if } f(x) \text{ is odd} \]
In our case, since \( f(x) = \sin^{11}(x) \) is odd and we are integrating from \(-1\) to \(1\):
\[ I = \int_{-1}^{1} \sin^{11}(x) \, dx = 0 \]

### Conclusion
Thus, the value of the integral \( \int_{-1}^{1} \sin^{11}(x) \, dx \) is:
\[ \boxed{0} \]