`int_0^(pi) cos^(101)x dx ` is equal to

Options

A. `pi/4`
B. `1/(102)`
C. `(pi/3)^(101)`
D. 0

Answer

To solve the integral \( I = \int_0^{\pi} \cos^{101}(x) \, dx \), we can use a property of definite integrals.

### Step-by-step Solution:

1. **Define the Integral**:
\[ I = \int_0^{\pi} \cos^{101}(x) \, dx \]

2. **Use the Property of Definite Integrals**:
We can use the property of integrals which states:
\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \]
Here, we set \( a = 0 \) and \( b = \pi \), so:
\[ I = \int_0^{\pi} \cos^{101}(\pi - x) \, dx \]

3. **Simplify the Integral**:
We know that:
\[ \cos(\pi - x) = -\cos(x) \]
Therefore:
\[ I = \int_0^{\pi} (-\cos(x))^{101} \, dx \]
Since 101 is an odd number, we have:
\[ I = \int_0^{\pi} -\cos^{101}(x) \, dx = -\int_0^{\pi} \cos^{101}(x) \, dx = -I \]

4. **Set Up the Equation**:
From the previous step, we have:
\[ I = -I \]

5. **Solve for \( I \)**:
Adding \( I \) to both sides gives:
\[ 2I = 0 \]
Thus:
\[ I = 0 \]

### Conclusion:
The value of the integral \( \int_0^{\pi} \cos^{101}(x) \, dx \) is:
\[ \boxed{0} \]