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.