What is the value of `int_(0)^(pi//2) cos^(8)dx` ?

Options

A. `(35pi)/(256)`
B. `(70)/(256)`
C. `(16)/(35)`
D. `(8pi)/(35)`

Explanation

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos^8 x \, dx \), we can use the properties of the gamma function and the known results for integrals of powers of sine and cosine.

### Step-by-Step Solution:

1. **Set Up the Integral**:
\[ I = \int_{0}^{\frac{\pi}{2}} \cos^8 x \, dx \]

2. **Use the Beta Function**:
The integral can be expressed in terms of the Beta function:
\[ I = \int_{0}^{\frac{\pi}{2}} \cos^{2n} x \, dx = \frac{1}{2} B\left(n + \frac{1}{2}, \frac{1}{2}\right) \]
where \( n = 4 \) (since \( 2n = 8 \)).

3. **Relate Beta Function to Gamma Function**:
The Beta function is related to the Gamma function:
\[ B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)} \]
Therefore,
\[ I = \frac{1}{2} \cdot \frac{\Gamma(4 + \frac{1}{2}) \Gamma(\frac{1}{2})}{\Gamma(4 + 1)} \]

4. **Calculate the Gamma Values**:
- \( \Gamma(\frac{1}{2}) = \sqrt{\pi} \)
- \( \Gamma(5) = 4! = 24 \)
- To calculate \( \Gamma(4 + \frac{1}{2}) \), we use:
\[ \Gamma(4 + \frac{1}{2}) = \frac{(7/2)!}{(7/2 - 1)(7/2 - 2)(7/2 - 3)(7/2 - 4)} \]
which can be simplified using the properties of the gamma function.

5. **Substituting the Values**:
After calculating \( \Gamma(4 + \frac{1}{2}) \), we can substitute back into the equation:
\[ I = \frac{1}{2} \cdot \frac{\Gamma(4 + \frac{1}{2}) \cdot \sqrt{\pi}}{24} \]

6. **Final Calculation**:
After performing the calculations, we find:
\[ I = \frac{35 \pi}{256} \]

### Final Answer:
Thus, the value of the integral \( \int_{0}^{\frac{\pi}{2}} \cos^8 x \, dx \) is:
\[ \boxed{\frac{35 \pi}{256}} \]