Integrate Using Secant Substitution

Integrate $\frac{dx}{\sqrt{4x^2 - 1}}$ using the secant substitution where $x = \frac{1}{2}\sec(\theta)$.

In this problem, we will explore a technique in integration known as trigonometric substitution, specifically using the secant substitution method. Trigonometric substitution is a strategic technique that takes advantage of the identity properties of trigonometric functions to simplify integrals. It is particularly effective when dealing with integrands that involve square roots of quadratic expressions, such as the one in this problem. By introducing a trigonometric function in place of a variable, we can transform the integrand into a form that is easier to integrate.

The secant substitution method involves making the substitution $x = \frac{1}{2} \sec(\theta)$, which draws a connection between the hyperbolic secant function and the trigonometric secant function. This particular substitution is useful because it aligns with the structure of the given integrand, specifically the $\sqrt{4x^2 - 1}$ term. The advantage here is that this substitution transforms the radical into a simple trigonometric identity, which can be further simplified by manipulating trigonometric identities and equations.

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