Trigonometric substitutions are used in integration primarily for what reason?

Trigonometric substitutions are primarily utilized in integration to simplify integrals that involve square root expressions, particularly those containing terms of the form √(a² - x²), √(x² - a²), or √(a² + x²). By making a suitable substitution, such as x = a sin(θ) or x = a tan(θ), the square root often transforms into a trigonometric function, which is generally easier to integrate.

This transformation leverages the inherent relationships and identities present in trigonometric functions, allowing for a straightforward integration process. For example, applying the substitution can convert expressions into forms where standard integral results or formulas can be employed conveniently, leading to a more manageable problem.

In contrast, other options don't capture the primary essence of trigonometric substitutions effectively. Making the integral more complicated does not align with the purpose of these substitutions. While converting algebraic functions into trigonometric ones is part of the process, the ultimate aim is to facilitate the simplification of the integration rather than a mere change in function type. Evaluating definite integrals quickly is secondary; the main goal is the simplification through substitutions that transform square roots into more tractable forms for integration.