The value of `int_(0)^(pi) cos^(11)xdx`, is

Options

A. `pi`
B. `11pi`
C. 0
D. `-11pi`

Explanation

To solve the integral \( I = \int_{0}^{\pi} \cos^{11}(x) \, dx \), we can use a symmetry property of definite integrals. Here’s a step-by-step solution:

### Step 1: Define the Integral
Let:
\[ I = \int_{0}^{\pi} \cos^{11}(x) \, dx \]

### Step 2: Use the Property of Definite Integrals
We can use the property that states:
\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]
In our case, \( a = 0 \) and \( b = \pi \). Thus, we have:
\[ I = \int_{0}^{\pi} \cos^{11}(\pi - x) \, dx \]

### Step 3: Simplify the Integral
Using the trigonometric identity \( \cos(\pi - x) = -\cos(x) \), we can rewrite the integral:
\[ I = \int_{0}^{\pi} (-\cos(x))^{11} \, dx \]
Since \( (-1)^{11} = -1 \), this simplifies to:
\[ I = -\int_{0}^{\pi} \cos^{11}(x) \, dx = -I \]

### Step 4: Solve for \( I \)
Now we have:
\[ I = -I \]
Adding \( I \) to both sides gives:
\[ 2I = 0 \]
Thus:
\[ I = 0 \]

### Final Answer
The value of the integral \( \int_{0}^{\pi} \cos^{11}(x) \, dx \) is:
\[ \boxed{0} \]