To evaluate the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos^9 x \, dx \), we can use the reduction formula for integrals of the form \( \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx \).
### Step-by-step Solution:
1. **Identify the Integral**:
We want to evaluate \( I = \int_{0}^{\frac{\pi}{2}} \cos^9 x \, dx \).
2. **Use the Reduction Formula**:
The reduction formula for \( \int_{0}^{\frac{\pi}{2}} \cos^n x \, dx \) is given by:
\[
\int_{0}^{\frac{\pi}{2}} \cos^n x \, dx = \frac{n-1}{n} \int_{0}^{\frac{\pi}{2}} \cos^{n-2} x \, dx
\]
We will apply this formula to our integral.
3. **Apply the Reduction Formula**:
For \( n = 9 \):
\[
I = \frac{8}{9} \int_{0}^{\frac{\pi}{2}} \cos^7 x \, dx
\]
Let \( I_7 = \int_{0}^{\frac{\pi}{2}} \cos^7 x \, dx \).
4. **Continue Applying the Reduction Formula**:
Now we need to evaluate \( I_7 \):
\[
I_7 = \frac{6}{7} \int_{0}^{\frac{\pi}{2}} \cos^5 x \, dx
\]
Let \( I_5 = \int_{0}^{\frac{\pi}{2}} \cos^5 x \, dx \).
5. **Repeat the Process**:
Continuing this process:
\[
I_5 = \frac{4}{5} \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx
\]
Let \( I_3 = \int_{0}^{\frac{\pi}{2}} \cos^3 x \, dx \).
6. **Final Step**:
\[
I_3 = \frac{2}{3} \int_{0}^{\frac{\pi}{2}} \cos x \, dx
\]
Since \( \int_{0}^{\frac{\pi}{2}} \cos x \, dx = 1 \), we have:
\[
I_3 = \frac{2}{3} \cdot 1 = \frac{2}{3}
\]
7. **Back Substitute**:
Now substituting back:
\[
I_5 = \frac{4}{5} I_3 = \frac{4}{5} \cdot \frac{2}{3} = \frac{8}{15}
\]
\[
I_7 = \frac{6}{7} I_5 = \frac{6}{7} \cdot \frac{8}{15} = \frac{48}{105}
\]
\[
I = \frac{8}{9} I_7 = \frac{8}{9} \cdot \frac{48}{105} = \frac{384}{945}
\]
8. **Simplify the Result**:
Finally, we can simplify \( \frac{384}{945} \):
\[
\frac{384}{945} = \frac{128}{315}
\]
Thus, the value of the integral \( I = \int_{0}^{\frac{\pi}{2}} \cos^9 x \, dx \) is \( \frac{128}{315} \).
