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Trigonometric substitution is a technique used in integral calculus to simplify certain integrals. It's especially useful when dealing with expressions involving radicals or quadratic forms. By substituting trigonometric functions for variables, you can transform a complex expression into a simpler form. This method works well when the expression fits the formats:

• \(a^2 - x^2\)
• \(x^2 - a^2\)
• \(x^2 + a^2\)

In the exercise, the expression in the denominator is transformed into one of these forms through completing the square, which reveals that trigonometric substitution using the arctangent identity is applicable. The method is basically identifying a substitution involving trigonometric identities to simplify integration. It's an elegant strategy that opens the door to integrating otherwise difficult expressions by relying on familiar trigonometric identities.

Completing the square transforms a quadratic expression into a perfect square trinomial. This method makes it easier to integrate functions involving quadratic expressions. The process involves:

• Identifying the quadratic expression, like \(x^2 + 4x + 13\).
• Taking half of the linear coefficient \(b\) of \(x\), squaring it, and adjusting the expression.
• For \(x^2 + 4x + 13\), take half of 4, which is 2, and square it to get 4.
• Rewrite as \((x+2)^2 + 9\) by adding and subtracting the square.

This makes integration simpler by framing the quadratic expression in a usable format for further steps, such as applying trigonometric substitution or integrating directly using specific formulas. Completing the square is a valuable tool in making complex expressions more manageable for integral calculations.

The arctangent integral formula is a specific result used for integrating expressions of the form \(\int \frac{1}{a^2 + x^2} \, dx \). This is a classic integration problem involving the inverse tangent function, \(\tan^{-1}\). The formula is:
\[ \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left( \frac{x}{a} \right) + C \]This formula is particularly useful when the integrand resembles a sum of squares, as completing the square allows.
In the given exercise, once the expression \(x^2 + 4x + 13\) is rewritten after completing the square, it becomes \((x+2)^2 + 9\). Setting \(a = 3\) and \(u = x+2\), the integral fits the form required for the arctangent integral formula. Applying this formula, and then back-substituting to replace any intermediate variable substitutions, yields the integral's solution. Understanding the arctangent integral formula is essential for tackling such calculus problems efficiently.