Completing the Square with Trigonometric Substitution

Complete the square for the expression $x^2 + 2x$ and then perform a trigonometric substitution.

Completing the square is a powerful algebraic technique that's often used to transform quadratic expressions into a form that is easier to analyze or integrate. This method involves manipulating the quadratic expression so that it resembles a perfect square trinomial, which is an expression of the form $(x + a)^2 + b$. By rewriting the quadratic in this manner, it becomes simpler to evaluate or integrate depending on the context of the problem.

In calculus, completing the square frequently acts as a precursor to trigonometric substitution, a technique applied to simplify integrals involving square roots. For functions that include expressions like $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$, trigonometric substitution allows us to exploit trigonometric identities to simplify these roots. By substituting a trigonometric function for $x$, the integral often becomes one that is much easier to solve because it reduces the problem to a standard trigonometric integral that you might be more familiar with.

For this problem, you'll start by completing the square on the quadratic expression $x^2 + 2x$. Once that is achieved, a trigonometric substitution can help transform the resulting expression into a more manageable form for integration or further analysis. This dual approach highlights the interplay between algebraic manipulation and trigonometric identities in calculus, which is a recurring theme when solving complex integrals.

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