Evaluating Integral with Trigonometric Substitution

Evaluate $\displaystyle \int \frac{\sqrt{4 - x^2}}{x^2} \, dx$ from $\sqrt{2}$ to 2 using trigonometric substitution.

This problem involves evaluating a definite integral using the method of trigonometric substitution, a powerful tool for simplifying and solving integrals that contain square roots of quadratic expressions. Trigonometric substitution takes advantage of the identities in trigonometry that relate to squares, such as the Pythagorean identities, to transform an integral into a simpler form where trigonometric identities make the evaluation more straightforward.

This technique often involves substituting a trigonometric function for a variable so that the resulting integral is easier to handle. For instance, in expressions involving $a^2 - x^2$, a substitution like $x = a \sin(\theta)$ is used because it simplifies $\sqrt{a^2 - x^2}$ into a form that integrates more smoothly in terms of $\theta$. This substitution also transforms the limits of integration, which requires careful attention to ensure accuracy.

The strategy for solving this kind of problem generally includes writing the integral in terms of the trigonometric identity, changing the variable and limits of integration, simplifying using trigonometric identities, and then integrating the resulting expression. Trigonometric substitution not only requires knowledge of integration techniques but also a solid understanding of trigonometric identities and their applications in calculus. Integrals like this often serve as an excellent exercise for applying trigonometric concepts in a calculus setting, reinforcing both algebraic manipulation and conceptual understanding of integrals involving square roots.

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