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The problem is asking to evaluate the integral using trigonometric substitution. The integral is:

\[
\int \frac{dx}{x^4 \sqrt{9 - x^2}}
\]

### Solution Steps:

1. **Identify the Trigonometric Substitution**:
   Notice that the integrand contains \(\sqrt{9 - x^2}\). This suggests using the substitution \(x = 3\sin\theta\), because \(\sqrt{9 - x^2}\) will simplify using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\).

   Set \(x = 3\sin\theta\), so \(dx = 3\cos\theta \, d\theta\).

2. **Substitute into the Integral**:
   Substituting \(x = 3\sin\theta\) and \(dx = 3\cos\theta \, d\theta\) into the integral:

   \[
   \int \frac{3\cos\theta \, d\theta}{(3\sin\theta)^4 \sqrt{9 - (3\sin\theta)^2}}
   \]

   Simplifying further:

   \[
   = \int \frac{3\cos\theta \, d\theta}{81\sin^4\theta \cdot \sqrt{9(1 - \sin^2\theta)}}
   \]

   Recall that \(\sqrt{1 - \sin^2\theta} = \cos\theta\), so:

   \[
   = \int \frac{3\cos\theta \, d\theta}{81\sin^4\theta \cdot 3\cos\theta}
   \]

   This simplifies to:

   \[
   = \frac{1}{81} \int \frac{d\theta}{\sin^4\theta}
   \]

3. **Simplify the Trigonometric Integral**:
   The integral \(\int \frac{d\theta}{\sin^4\theta}\) can be rewritten using the identity \(\csc\theta = \frac{1}{\sin\theta}\):

   \[
   = \frac{1}{81} \int \csc^4\theta \, d\theta
   \]

   This integral is typically solved using reduction formulas or by recognizing standard integrals. The reduction formula for \(\int \csc^n \theta \, d\theta\) is complex, so instead, we'll recognize a known integral:

   \[
   \int \csc^4 \theta \, d\theta = -\frac{\cos\theta \csc^2\theta}{3} - \frac{2\theta}{3} + C
   \]

   Thus:

   \[
   \frac{1}{81} \left(-\frac{\cos\theta \csc^2\theta}{3} - \frac{2\theta}{3} \right) + C
   \]

4. **Back-Substitute \(x\)**:
   Recall \(x = 3\sin\theta\), so \(\sin\theta = \frac{x}{3}\). Therefore, \(\theta = \arcsin\left(\frac{x}{3}\right)\).

   Also, \(\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \frac{x^2}{9}}\).

   Substituting back into the expression:

   \[
   -\frac{1}{243} \left(\sqrt{1 - \frac{x^2}{9}} \cdot \frac{9}{x^2} \right) - \frac{2}{243}\arcsin\left(\frac{x}{3}\right) + C
   \]

   Simplify the expression to get the final answer.

Would you like me to go into more detail on any part of the solution, or do you have any specific questions?

### Related Questions:
1. How do you evaluate integrals involving \(\sqrt{a^2 - x^2}\) in general?
2. What are the standard trigonometric substitution identities and when are they applied?
3. How is the integral of \(\csc^n \theta\) derived?
4. Can trigonometric substitution be used for integrals involving \(\sqrt{x^2 + a^2}\)?
5. What is the geometric interpretation of trigonometric substitution in integrals?

### Tip:
When using trigonometric substitution, always double-check the identity you are using, and remember to convert back to the original variable at the end.

Evaluate the following integral using trigonometric substitution: ∫ dx / (x^4 * sqrt(9 - x^2))

This guide explains how to evaluate the integral ∫ dx / (x^4 * sqrt(9 - x^2)) using trigonometric substitution. Learn about the substitution method, key formulas, and how to simplify complex trigonometric integrals. Suitable for students in advanced calculus courses.

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