Integral of Square Root of Quadratic Over x

Evaluate the integral of $\displaystyle \frac{\sqrt{25x^2 - 4}}{x}$ with respect to $x$.

When tasked with evaluating integrals, particularly those involving a square root of a quadratic expression, a strategic approach is necessary. In this scenario, the integral involves the square root of $25x^2 - 4$ divided by $x$. Such problems typically hint at benefiting from a trigonometric substitution strategy because the integrand resembles forms amenable to this technique. Trigonometric substitution is a powerful tool, designed to simplify the square root of quadratic expressions by substituting variables in terms of trigonometric functions, effectively transforming the integral into a more manageable trigonometric integral or other familiar integral forms.

For this specific integral, recognize the form $a^2 - b^2x^2$ under the square root, which suggests a trigonometric substitution such as $x = \frac{a}{b}\sec(\theta)$. Such a substitution utilizes the identity $\sec^2(\theta) - 1 = \tan^2(\theta)$, which can help sidestep the complexity of square roots. By adopting this substitution, the integration process can lead us into handling trigonometric integrals, which involve standard forms and can often be evaluated with known results. The strategy involves careful manipulation and algebraic transformation of the integral into a trigonometric form, followed by application of integration techniques for trigonometric functions, and finally substituting back to the original variable, $x$. This multi-step process underscores the importance of selecting appropriate substitution techniques based on the structure of the integrand and emphasizes recognizing patterns that align with standard integral forms.

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