91Ó°ÊÓ

Trigonometric substitution is a powerful technique used to evaluate certain integrals, especially when the integral involves expressions like \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \). In the given problem, we are dealing with \( \sqrt{1 - 9x^2} \). By substituting \( x = \frac{1}{3} \sin{\theta} \), we exploit the Pythagorean identity \( 1 - \sin^2{\theta} = \cos^2{\theta} \). This allows us to transform the square root: \( \sqrt{1 - 9x^2} \) becomes the simpler form \( \cos{\theta} \).

With this substitution, the integral simplifies significantly, turning a complex expression into a basic trigonometric function. One common criterion for choosing substitutions is to match the integrand to a standard integral that simplifies our calculations. Therefore, by efficiently applying these trigonometric identities, we can handle otherwise challenging integrals with greater ease.

The change of variables is akin to a creative shortcut in integration. It involves replacing a variable with another expression that simplifies the integral. In the context of indefinite integrals, like our example, the intuitive step is finding a substitution that eases calculation.

For this specific problem, we used \( x = \frac{1}{3} \sin{\theta} \), which transforms \( dx \) into \( \frac{1}{3} \cos{\theta} d\theta \). The reason behind this choice is to work with a simpler integral rooted in trigonometric identities. Once the original variables are changed to others that simplify the mathematical process, integrating becomes more straightforward. This prowess in simplifying complex expressions often illuminates the path to the integral's solution.

Chain rule differentiation is essential when checking the correctness of indefinite integral solutions. It states that \( \frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x) \). In the context of integration, once the antiderivative is found, we differentiate to verify our result matches the integrand.

In this particular problem, after integrating to obtain \( \frac{1}{3} \arcsin(3x) + C \), differentiating it should give us the original integrand. Utilizing the chain rule, we proceed by differentiating \( \arcsin(3x) \). The external derivative of \( \arcsin(u) \) is \( \frac{1}{\sqrt{1-u^2}} \) and the derivative of \( 3x \) leads to a factor of 3, confirming the integrand perfectly. This derivative consistency check underscores the integral's validity and highlights the chain rule as an invaluable tool in calculus for ensuring accuracy.