Evaluate :
`int_(0)^(pi//2)cos^(9)xdx`

Answer

To evaluate the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos^9 x \, dx \), we can use Wallis's formula for integrals of the form \( \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx \) where \( n \) is an odd integer.

### Step-by-step Solution:

1. **Identify the formula**:
Wallis's formula states that for an odd integer \( n \):
\[ \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx = \frac{n(n-1)(n-2)\cdots(3)(1)}{n(n-1)(n-2)\cdots(2)(1)} \cdot \frac{\pi}{2} \]

2. **Determine \( n \)**:
In our case, \( n = 9 \).

3. **Calculate the numerator**:
The numerator is given by:
\[ 9 \cdot 7 \cdot 5 \cdot 3 \cdot 1 \]
We can compute this step-by-step:
- \( 9 \cdot 7 = 63 \)
- \( 63 \cdot 5 = 315 \)
- \( 315 \cdot 3 = 945 \)
- \( 945 \cdot 1 = 945 \)

4. **Calculate the denominator**:
The denominator is given by:
\[ 8 \cdot 6 \cdot 4 \cdot 2 \]
We can compute this step-by-step:
- \( 8 \cdot 6 = 48 \)
- \( 48 \cdot 4 = 192 \)
- \( 192 \cdot 2 = 384 \)

5. **Combine the results**:
Now, we can substitute the values into the formula:
\[ I = \frac{945}{384} \cdot \frac{\pi}{2} \]

6. **Simplify the expression**:
To simplify:
\[ I = \frac{945 \cdot \pi}{768} \]

7. **Final answer**:
The value of the integral is:
\[ I = \frac{945 \pi}{768} \]

### Final Result:
Thus, the evaluated integral is:
\[ \int_{0}^{\frac{\pi}{2}} \cos^9 x \, dx = \frac{945 \pi}{768} \]