91Ó°ÊÓ

Trigonometric substitution is a powerful technique used to simplify integrals, especially when they involve expressions like \( \sqrt{a^2 - x^2} \). The idea is to use a trigonometric identity to transform an integral into a simpler form.
For the integral \( \int \frac{x^2}{\sqrt{9-x^2}} \, dx \), the substitution \( x = 3\sin\theta \) is utilized because it simplifies the square root. This substitution transforms the radical \( \sqrt{9-x^2} \) into \( 3\cos\theta \), reducing the complexity of the integral.
To execute this substitution, remember that:

• \( dx = 3\cos\theta \, d\theta \)
• The substitution \( x = 3\sin\theta \) aligns with the identity \( \sin^2\theta + \cos^2\theta = 1 \)

By choosing substitutions wisely, we can convert difficult integrals into forms that are much easier to solve.

Trigonometric identities play a central role in simplifying integrals. In our exercise, we encounter the integral \( \int 9\sin^2\theta \, d\theta \), which can be tackled using a well-known trigonometric identity.
The identity \( \sin^2\theta = \frac{1-\cos(2\theta)}{2} \) is particularly useful as it helps break down the problem into more manageable parts, making integration simple. Applying this identity transforms the integral into:

• \( \int 9 \left( \frac{1-\cos(2\theta)}{2} \right) \, d\theta = \frac{9}{2} \int (1-\cos(2\theta)) \, d\theta \)

Trigonometric identities like these help decompose complex trigonometric expressions into linear forms, allowing for straightforward integration of each term.

Inverse trigonometric functions are essential when reversing a trigonometric substitution to obtain the integral's result in terms of the original variable \( x \).
In our problem, after integration, we reach an expression featuring \( \theta \) that needs to be transformed back to \( x \). Since \( \theta = \arcsin\left( \frac{x}{3} \right) \), this substitution helps express our answer in terms of inverse trigonometric functions.
Using the relations:

• \( \sin\theta = \frac{x}{3} \) implies \( \theta = \arcsin\left( \frac{x}{3} \right) \)
• \( \sin(2\theta) = 2\sin\theta\cos\theta \) becomes \( \frac{2x\sqrt{9-x^2}}{9} \)

These functions bridge the step from trigonometric identities and rules back to a real-world variable, providing a coherent final solution of \[ \frac{9}{2} \arcsin\left( \frac{x}{3} \right) - \frac{x\sqrt{9-x^2}}{2} + C \]. Inverse trigonometric functions allow the conversion of trigonometric results into understandable, practical terms.