Integrating with Trigonometric Substitution using Secant Function

$\displaystyle \int \frac{\sqrt{x^2-25}}{x} \, dx$ with $x = 5 \sec \theta$

Trigonometric substitution is a powerful technique in integration that is particularly useful for integrals involving square roots of quadratic expressions. In this problem, the expression under the square root suggests a trigonometric identity involving secant, since it involves the form a squared minus x squared. Substituting x equals 5 secant theta takes advantage of the identity that secant squared theta minus one equals tangent squared theta, allowing the square root to be simplified significantly.

Once the substitution is made, the differential dx must also be converted into a form that accounts for theta. This often involves finding dx in terms of d theta, which in this problem results in a combination of secant and tangent functions. After substitution, the integral transforms into an order that is easier to handle, frequently requiring a secondary technique like trigonometric identities or another form of integration technique to solve completely.

This particular problem highlights the utility of trigonometric substitution as a strategic approach enhancing the process of finding closed forms for complex integral expressions. Understanding when and how to employ this strategy is crucial for solving integrals that initially appear daunting due to the presence of square roots of quadratics.

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