Evaluate the Integral Involving Square Root

Evaluate the integral $\displaystyle \int_{1}^{2} \frac{\sqrt{x^2 - 1}}{x} \, dx$.

This problem requires evaluating a definite integral that involves a square root expression in the numerator and a rational function form. To approach this problem, you can utilize trigonometric substitution, a technique often employed when dealing with integrals that have expressions of the form square root of a squared minus x squared, x squared minus a squared, or a squared plus x squared.

The substitution helps to simplify the square root expression by relating it to a trigonometric identity, which consequently converts the integral into a more manageable form that's easier to evaluate. In this case, by choosing an appropriate trigonometric substitution, the integral can be transformed into one involving a simple trigonometric function. This method not only simplifies the square root but also deals with the rational function present in the integrand.

Understanding when and how to apply trigonometric substitution is crucial for tackling integrals of this nature, and mastering this technique is a valuable skill in calculus. It enables you to recognize patterns and choose substitutions that reduce complex integrals to basic ones that can be solved using standard antiderivatives or further integration techniques. This problem also highlights the importance of evaluating definite integrals, paying close attention to the limits of integration after substituting and simplifying.

Related Problems