91Ó°ÊÓ

Trigonometric identities are equations that define the relationships between various trigonometric functions. They play a crucial role in solving many mathematical problems, especially when working with integrals. In the context of the given exercise, the identity used is \( \sec^2(\theta) = 1 + \tan^2(\theta) \). This identity simplifies the integrand \( \sqrt{\sec^2 \theta - 1} \) by recognizing that it can be rewritten in terms of tangent: \( \sqrt{\tan^2 \theta} \).

Understanding these fundamental relationships allows us to transform complex expressions into simpler forms that we can integrate more easily. It is also essential to recognize the range over which the function is being integrated to determine the correct sign of the trigonometric function, just as the exercise demonstrated with tangent being positive in the given interval from \( -\frac{x}{3} \) to \( \frac{\pi}{3} \).

An antiderivative of a function is another function whose derivative gives the original function. Finding an antiderivative, also known as integration, is a fundamental process in calculus. In our exercise, the antiderivative of \( \tan(\theta) \) is found to be \( \ln|\sec(\theta)| + C \), where \( C \) represents the constant of integration for indefinite integrals. This step is vital because it sets up the problem to apply the Fundamental Theorem of Calculus for definite integrals.

Finding the antiderivative requires a combination of knowledge of basic derivative formulas and some ingenuity in recognizing patterns and potential substitutions. As such, remembering key derivatives, like that of \( \ln|\sec(\theta)| \) in relation to \( \tan(\theta) \), is exceptionally useful for solving integration problems efficiently.

The Fundamental Theorem of Calculus (FTC) bridges the concept of derivative and integral, providing a way to evaluate definite integrals when the antiderivative of a function is known. It states that if \( F \) is an antiderivative of \( f \) on an interval \( [a, b] \) then \( \int_{a}^{b} f(x) dx = F(b) - F(a) \).

In the given problem, once the antiderivative of the integrand \( \tan(\theta) \) was identified as \( \ln|\sec(\theta)| \), the FTC allowed us to evaluate the definite integral by calculating the difference \( F(\pi/3) - F(-x/3) \). This is where we computed \( \ln|\sec(\pi/3)| - \ln|\sec(-x/3)| \), providing the final evaluated integral.

Comprehending the FTC is critical for anyone studying calculus as it grants the power to transition from anti-differentials to concrete values of area under a curve, which is often the goal when working with definite integrals.