Convergence of Integral with Square Root in Denominator

Determine whether the integral from 2 to 3 of $\displaystyle \int_{2}^{3} \frac{1}{\sqrt{3 - x}} \, dx$ is convergent or divergent and evaluate it if it is convergent.

This problem challenges one to determine the convergence of a definite integral where the integrand contains a square root that becomes undefined at the endpoint of the interval. This classifies the integral as improper due to the behavior near the boundary of integration. To solve this type of problem, one can rewrite the integral in a form that highlights its convergence properties relative to specific types of integrals. A common approach is to compare it to a known integral form, such as p-integrals, to assess convergence.

When dealing with integrals of these forms, focusing on the limits as the variable approaches the problematic point will be crucial. Evaluating whether the integral is convergent involves determining whether limits that replace problematic points yield a finite value or become infinite. If the integral proves convergent, one must then execute an evaluation which might involve substitution methods to simplify the integral into a more straightforward, elementary form.

Understanding the theory behind improper integrals, including familiarity with divergence and convergence techniques, is essential. Such problems often involve a balance of algebraic manipulation and conceptual understanding of limit behavior. This specific problem is exemplary for exploring the complexities of calculus regarding integrals with undefined points in their domain, thus broadening one's skill in evaluating integral convergence.

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