91Ó°ÊÓ

Completing the square is a technique used to turn a quadratic expression into a perfect square trinomial, which makes it easier to work with, especially in the context of integration. The original quadratic expression in this problem is \(5 - 4x - x^2\).
This expression can be rearranged as \(- (x^2 + 4x - 5)\), which is more suitable for the completing the square process. To complete the square:\

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• First, focus on the terms \(x^2 + 4x\). Here, take half of the coefficient of \(x\), which is 4, giving you 2.\
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• Square this number, resulting in 4.\
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• Add and subtract this square inside the quadratic expression. This step allows us to write it as \((x+2)^2 - 4\).\
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Thus, \(5 - 4x - x^2\) can be rewritten as \(9 - (x+2)^2\). This completed square form simplifies the process of integration, often leading to the use of trigonometric substitution.

Trigonometric substitution is a method utilized to solve integrals involving square roots of quadratic expressions. In the current integral, after completing the square, we arrived at \(\int \sqrt{9 - (x+2)^2} \, dx\).
This expression is similar to the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). For our expression:\

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• We let \(u = x + 2\), transforming the integral to \(\int \sqrt{9 - u^2} \, du\).\
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• Here, \(9\) can be thought of as \(3^2\), suggesting a \(\sin\) substitution.\
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Trigonometric substitutions often involve letting \(u = a \sin(\theta)\) or \(u = a \cos(\theta)\). By recognizing \(\int \sqrt{a^2 - u^2} \, du\), we can utilize known integration formulas from tables, simplifying our work greatly. This type of integral is standard and can often be found in integral tables.

Evaluating an integral involves finding the function whose derivative results in the original integrand. With the substitution made earlier, the integral \(\int \sqrt{9 - u^2} \, du\) is transformed into a classic form.
We recognize this matches the form \(\int \sqrt{a^2 - u^2} \, du\), with \(a = 3\). The evaluation involves:

• Using a standard formula: \(\frac{1}{2} (u \sqrt{a^2 - u^2} + a^2 \sin^{-1}\left(\frac{u}{a}\right)) + C\).\
• Substituting \(a = 3\) into this formula.\

This gives us \(\frac{1}{2} (u \sqrt{9-u^2} + 9 \sin^{-1}(\frac{u}{3})) + C\). Finally, substitute back \(u = x + 2\) to return to our original variable. Thus, the evaluated integral is \(\frac{1}{2} ((x+2) \sqrt{5-4x-x^2} + 9 \sin^{-1}(\frac{x+2}{3})) + C\). This process ensures the integral is expressed clearly in terms of the original variable \(x\), providing the evaluated result of the integral.