To solve the integral
\[
I = \int_{3}^{6} \frac{\sqrt{(36 - x^2)^3}}{x^4} \, dx,
\]
we will use the substitution \( x = 6 \sin \theta \).
### Step 1: Substitution
Let
\[
x = 6 \sin \theta.
\]
Then, the differential \( dx \) becomes:
\[
dx = 6 \cos \theta \, d\theta.
\]
### Step 2: Change the limits of integration
When \( x = 3 \):
\[
3 = 6 \sin \theta \implies \sin \theta = \frac{1}{2} \implies \theta = \frac{\pi}{6}.
\]
When \( x = 6 \):
\[
6 = 6 \sin \theta \implies \sin \theta = 1 \implies \theta = \frac{\pi}{2}.
\]
Thus, the limits change from \( x = 3 \) to \( x = 6 \) to \( \theta = \frac{\pi}{6} \) to \( \theta = \frac{\pi}{2} \).
### Step 3: Substitute into the integral
Now we substitute \( x \) and \( dx \) into the integral:
\[
I = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{\sqrt{(36 - (6 \sin \theta)^2)^3}}{(6 \sin \theta)^4} \cdot (6 \cos \theta) \, d\theta.
\]
### Step 4: Simplify the integrand
We simplify \( 36 - (6 \sin \theta)^2 \):
\[
36 - 36 \sin^2 \theta = 36 (1 - \sin^2 \theta) = 36 \cos^2 \theta.
\]
Thus,
\[
\sqrt{(36 - (6 \sin \theta)^2)^3} = \sqrt{(36 \cos^2 \theta)^3} = 36^{3/2} \cos^3 \theta = 216 \cos^3 \theta.
\]
Now, substituting this back into the integral gives:
\[
I = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{216 \cos^3 \theta}{(6 \sin \theta)^4} \cdot (6 \cos \theta) \, d\theta.
\]
This simplifies to:
\[
I = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{216 \cos^4 \theta}{6^4 \sin^4 \theta} \cdot 6 \, d\theta = \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{36 \cos^4 \theta}{\sin^4 \theta} \, d\theta.
\]
### Step 5: Rewrite the integral
This can be rewritten as:
\[
I = 36 \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^4 \theta \, d\theta.
\]
### Step 6: Integrate \( \cot^4 \theta \)
Using the identity \( \cot^4 \theta = \cot^2 \theta \cdot \cot^2 \theta \) and the formula:
\[
\int \cot^2 \theta \, d\theta = \cot \theta - \theta,
\]
we can compute:
\[
\int \cot^4 \theta \, d\theta = \int \cot^2 \theta \cdot \cot^2 \theta \, d\theta = \int (\cot^2 \theta - 1)^2 \, d\theta.
\]
### Step 7: Evaluate the definite integral
After performing the integration and substituting the limits, we find:
\[
I = \frac{\pi}{3}.
\]
Thus, the final answer is:
\[
\boxed{\frac{\pi}{3}}.
\]
