Evaulate the integral by trigonometric substitution \[ \int \frac{d x}{\left(4 x^{2}-9\right)^{3 / 2}} \]

Step 1 ：Let's use the trigonometric substitution method to solve this integral. The given integral is in the form of \(\int \frac{d x}{\left(a^2 x^{2}-b^2\right)^{n}}\), where a^2 = 4, b^2 = 9, and n = 3/2.

Step 2 ：Use the substitution x = (b/a) * cot(θ) and dx = -(b/a) * csc^2(θ) * dθ. In this case, a = 2, b = 3, and x = \(\frac{3}{2}\cot(θ)\).

Step 3 ：Calculate dx: dx = -\(\frac{3}{2}\) * csc^2(θ) * dθ.

Step 4 ：Rewrite the integral in terms of θ: \(\int \frac{-\frac{3}{2} \csc^2(θ) dθ}{\left(4 \left(\frac{3}{2} \cot(θ)\right)^{2}-9\right)^{3 / 2}}\)

Step 5 ：Simplify the integral: \(-1.5\int_{0}^{\frac{\pi}{2}} \frac{\csc^2(θ)}{(4\cot^2(θ) - 9)^{3/2}} dθ\)

Step 6 ：\(\boxed{-1.5\int_{0}^{\frac{\pi}{2}} \frac{\csc^2(θ)}{(4\cot^2(θ) - 9)^{3/2}} dθ}\)