Trigonometric Substitution with Completed Square

Evaluate $\displaystyle \int \frac{x}{\sqrt{(x+5)^2 + 4}} \, dx$ using trigonometric substitution after completing the square.

In calculus, evaluating integrals using trigonometric substitution is a powerful technique that simplifies complex expressions, especially when they involve square roots. The process often begins by completing the square in the integrand, which reshapes the expression under the square root into a more recognizably trigonometric form. This allows us to substitute a trigonometric function that simplifies the integral into a basic form that can be more easily solved.

In the specific problem of evaluating the integral of x over the square root of the quantity (x+5) squared plus 4, completing the square transforms this expression into one where trigonometric substitution is more straightforward. Typically, such substitutions involve using identities like the tangent or sine substitution, depending on the form achieved post-completion of square. For example, if you complete the square and have an expression that looks like the square root of a squared plus u squared, using a substitution involving tangent is effective.

The key to mastering these integrals is recognizing patterns and practicing manipulation techniques to transform the integrand into a suitable form for substitution. Understanding the underlying geometry and trigonometric identities involved provides deeper insights, helping students intuitively grasp not just the 'how', but the 'why' behind these substitutions. This approach promotes a broader comprehension that is not only useful for solving exam problems but also for applying these methods in various scientific contexts.

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