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

Options

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

Explanation

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

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int_{0}^{\pi} \cos^{11} x \, dx \]

### Step 2: Use the Property of Cosine
We can use the substitution \( x = \pi - t \). Then, \( dx = -dt \). Changing the limits accordingly:
- When \( x = 0 \), \( t = \pi \)
- When \( x = \pi \), \( t = 0 \)

Thus, the integral becomes:
\[ I = \int_{\pi}^{0} \cos^{11}(\pi - t)(-dt) = \int_{0}^{\pi} \cos^{11} t \, dt \]

### Step 3: Simplify the Cosine Function
Using the property \( \cos(\pi - t) = -\cos(t) \), we have:
\[ \cos^{11}(\pi - t) = (-\cos t)^{11} = -\cos^{11} t \]

### Step 4: Combine the Integrals
Now substituting back into the integral:
\[ I = \int_{0}^{\pi} -\cos^{11} t \, dt \]
This gives us:
\[ I = -\int_{0}^{\pi} \cos^{11} t \, dt = -I \]

### Step 5: Solve for \( I \)
Adding \( I \) to both sides:
\[ I + I = 0 \implies 2I = 0 \implies I = 0 \]

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