Step 1: Understanding the Concept:
To integrate an expression of the form $\frac{1}{\sqrt{ax^2 + bx + c}}$, we complete the square for the quadratic expression under the square root.
Step 2: Formula Application:
Rearrange $-4x^2 + 9x$:
$$-4\left(x^2 - \frac{9}{4}x\right) = -4\left[\left(x - \frac{9}{8}\right)^2 - \frac{81}{64}\right] = 4\left[\left(\frac{9}{8}\right)^2 - \left(x - \frac{9}{8}\right)^2\right]$$
The integral becomes:
$$\int \frac{dx}{\sqrt{4 [ (\frac{9}{8})^2 - (x - \frac{9}{8})^2 ]}} = \frac{1}{2} \int \frac{dx}{\sqrt{(\frac{9}{8})^2 - (x - \frac{9}{8})^2}}$$
Step 3: Explanation:
Using the standard formula $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a})$:
$$\frac{1}{2} \sin^{-1} \left( \frac{x - 9/8}{9/8} \right) = \frac{1}{2} \sin^{-1} \left( \frac{8x - 9}{9} \right) + C$$
Step 4: Final Answer:
The correct option is (c).