A

Step 1: Evaluate the first integral:
We need to calculate \( \int_0^1 9x^8 \, dx \).
The anti-derivative of \( 9x^8 \) is \( \frac{9}{9} x^9 = x^9 \).
Now, applying the limits from 0 to 1:
\[ \int_0^1 9x^8 \, dx = [x^9]_0^1 = 1^9 - 0^9 = 1 - 0 = 1. \]

Step 2: Evaluate the second integral:
We need to calculate \( \int_0^{\frac{\pi}{2}} \cos x \, dx \).
The anti-derivative of \( \cos x \) is \( \sin x \).
Now, applying the limits from 0 to \( \frac{\pi}{2} \):
\[ \int_0^{\frac{\pi}{2}} \cos x \, dx = [\sin x]_0^{\frac{\pi}{2}} = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1. \]

Step 3: Combine both results:
The total value is:
\[ \int_0^1 9x^8 \, dx + \int_0^{\frac{\pi}{2}} \cos x \, dx = 1 + 1 = 2. \]

Therefore, the final answer is 2.