Solving an Integral Using Completing the Square and Trigonometric Substitution

Solve the integral of $\displaystyle \int \frac{\sqrt{x+2}^3(x^2+4x)}{(x+2)^{-3/2}} \, dx$ by completing the square and using a trigonometric substitution.

This problem requires you to solve the integral by utilizing two main techniques: completing the square and trigonometric substitution. These techniques often go hand in hand when dealing with integrals involving square roots or expressions that suggest trigonometric identities.

To approach this problem, first consider how completing the square can simplify the algebraic expression inside the integral. Completing the square is a method used to transform a quadratic expression into a perfect square trinomial, which can be advantageous in integration as it often leads to forms amenable to trigonometric identities. By rewriting the expression under the square root, you can set the foundation for a smoother substitution process.

Once the expression is prepared via completing the square, trigonometric substitution becomes a useful strategy, particularly when dealing with integrands that resemble trigonometric forms. Trigonometric substitutions are particularly effective because they leverage the identities and derivatives of trigonometric functions, simplifying complex algebra into more manageable forms and often transforming square roots into more "integrable" trigonometric expressions. By identifying patterns that resemble forms of cosine, sine, or tangent within the integrand, you can substitute the appropriate trigonometric expression and simplify the integration process. This dual-strategy approach not only simplifies the integration but enhances the understanding of how algebraic and trigonometric manipulation can be used synergistically in calculus.

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