The integral you've provided is:

$\int \frac{1}{x^2 \sqrt{a^2 - x^2}} \, dx$

To solve this using trigonometric substitution, we can use the following substitution for expressions of the form $a^2 - x^2$:

Step 1: Trigonometric Substitution

Let: $x = a \sin(\theta)$ Then, we differentiate: $dx = a \cos(\theta) \, d\theta$ Also, using this substitution, the expression $\sqrt{a^2 - x^2}$ simplifies as follows: $\sqrt{a^2 - x^2} = \sqrt{a^2 - a^2 \sin^2(\theta)} = a \cos(\theta)$

Step 2: Substituting into the Integral

Substitute $x = a \sin(\theta)$ and $dx = a \cos(\theta) \, d\theta$ into the integral: $\int \frac{1}{x^2 \sqrt{a^2 - x^2}} \, dx = \int \frac{1}{a^2 \sin^2(\theta) \cdot a \cos(\theta)} \cdot a \cos(\theta) \, d\theta$

Simplifying the terms: $= \int \frac{1}{a^3 \sin^2(\theta)} \cdot a \cos(\theta) \, d\theta$ $= \frac{1}{a^2} \int \frac{\cos(\theta)}{\sin^2(\theta)} \, d\theta$

Step 3: Simplifying the Integral

This integral can be rewritten using the identity $\csc(\theta) = \frac{1}{\sin(\theta)}$, so we have: $= \frac{1}{a^2} \int \csc^2(\theta) \, d\theta$

The integral of $\csc^2(\theta)$ is: $\int \csc^2(\theta) \, d\theta = -\cot(\theta)$

Step 4: Back Substitution

We now substitute back for $\theta$ using the original substitution $x = a \sin(\theta)$. From this, we know: $\sin(\theta) = \frac{x}{a}, \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\sqrt{a^2 - x^2}}{x}$

Thus, the solution to the integral is: $\frac{1}{a^2} \left( -\frac{\sqrt{a^2 - x^2}}{x} \right) + C$ Simplifying: $-\frac{\sqrt{a^2 - x^2}}{a^2 x} + C$

This is the final answer.

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Related Questions:

1. What are the common trigonometric substitutions for integrals involving $\sqrt{a^2 - x^2}$?
2. How can you approach solving integrals using different substitution methods?
3. What is the geometric interpretation of trigonometric substitutions in integrals?
4. How would the integral change if the integrand involved $\sqrt{x^2 - a^2}$?
5. Can you explain how the inverse trigonometric functions relate to these types of integrals?

Tip:

For integrals involving square roots of the form $a^2 - x^2$, $a^2 + x^2$, or $x^2 - a^2$, trigonometric substitutions like $x = a \sin(\theta)$, $x = a \tan(\theta)$, or $x = a \sec(\theta)$ are very useful tools.