91Ó°ÊÓ

In integral calculus, trigonometric substitution is a powerful technique used to simplify integrals involving square roots of quadratic expressions. It is particularly effective when dealing with functions containing terms like \( \sqrt{x^2 - 1} \).

In this technique, we replace a variable with a trigonometric function to take advantage of trigonometric identities. For the integral in our exercise, we set \( x = \sec(\theta) \) because it allows us to transform \( x \sqrt{x^2 - 1} \) into a form that is easier to handle: \( \sec(\theta) \cdot \tan(\theta) \).

This step-by-step process not only simplifies the integral but also changes its variable from \( x \) to \( \theta \), leading to a simpler integration process.

• Choose the appropriate trigonometric substitution for \( x \).
• Convert \( dx \) accordingly using the derivative of the substitution.
• Simplify the integral with the new expression.

Once the integral is simplified, we can proceed with an easier evaluation.

Inverse trigonometric functions are essential when dealing with integrals that involve trigonometric substitutions. Functions like \( \sec^{-1}(x) \) reverse the trigonometric function, giving us an angle when provided with a ratio.

In our given problem, the term \( \cos(\sec^{-1}(x)) \) might initially appear complex. However, by substituting \( x = \sec(\theta) \), we know that \( \theta = \sec^{-1}(x) \), thus making \( \cos(\theta) = \frac{1}{x} \). This substitution simplifies the integral and transforms it into a more straightforward calculus problem.

• Understand each inverse function and its domain.
• Remember the geometric interpretation, such as right triangles, which aids in understanding these inverse relationships.

Utilizing inverse trigonometric functions effectively turns challenging integrals into solvable ones by reducing complexity.

Definite integrals calculate the accumulated quantity, such as area under a curve, over a specific interval. In this particular problem, we focus on evaluating a definite integral through substitution.

Once we replace \( x \) with \( \sec(\theta) \), the limits of integration also change from values of \( x \) to angles in terms of \( \theta \).

New limits are crucial: when \( x = \frac{2}{\sqrt{3}} \), \( \theta = \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \), and for \( x = 2 \), \( \theta = \frac{\pi}{3} \).

• Transform both the integral and its limits.
• Evaluate the integral using the new limits, which often leads to simpler calculations.

After substitution and transformation, we evaluate the integral, ultimately calculating the area or another accumulated measure determined by the function over the defined interval.