Evaluate the Integral Using Trigonometric Substitution2

Evaluate $\displaystyle \int \frac{\sqrt{2x^2 - 1}}{x} \, dx$ using trigonometric substitution.

This problem involves evaluating an integral by using the technique of trigonometric substitution. Trigonometric substitution is a powerful method often used to simplify integrals involving square roots of polynomial expressions. By substituting a trigonometric function for the variable, it transforms the integral into one that involves trigonometric identities, which are generally easier to handle. This method typically requires a good understanding of trigonometric identities and inverse trigonometric functions.

In this problem, the expression under the square root, $2x^2 - 1$, fits the form that suggests a trigonometric substitution could simplify the integration. Specifically, for expressions involving the square root of forms like $a^2 - x^2$, $x^2 - a^2$, or $x^2 + a^2$, it is common to use substitutions that involve sine, secant, or tangent functions respectively. In this case, recognizing the form and choosing an appropriate substitution is key. After substitution, you obtain an integral in terms of the trigonometric function, which can be solved using standard integration techniques.

Understanding the process of reverting back to the original variable is also crucial. Once the integration with respect to the trigonometric variable is complete, inverse trigonometric functions or identities are used to substitute back and express the antiderivative in terms of the original variable. This technique, although technical, provides robust tools for handling integrals that otherwise appear complex or unmanageable.

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