Question:

Step 1: Simplify $f(x)$. Use $1+\tan^2 x = \sec^2 x$ and $1+\cot^2 x = \csc^2 x$: $$f(x) = \frac{3}{\sec^2 x} + \frac{9}{\csc^2 x} = 3\cos^2 x + 9\sin^2 x$$ $$= 3\cos^2 x + 9(1-\cos^2 x) = 9 - 6\cos^2 x = 9 - 3(1+\cos 2x) = 6 - 3\cos 2x$$ Step 2: Determine the domain. $f(x)$ is undefined when $\tan x$ or $\cot x$ is undefined, i.e., $x \neq n\pi/2$ for $n \in \mathbb{Z}$. At these excluded points, $\cos 2x = \pm 1$. Step 3: Find the range. $\cos 2x \in (-1, 1)$ (excluding $\pm 1$ since $x \neq n\pi/2$). $$f(x) = 6 - 3\cos 2x \in (6-3, 6+3) = (3, 9)$$ Step 4: Count integers in $(3, 9)$: integers are $4, 5, 6, 7, 8$ → **5 integers**. Option (b) is correct. Step 5: Sum of integers $= 4+5+6+7+8 = 30$. Option (c) is correct.