Evaluate Integral Using Trigonometric Substitution23

Evaluate $\displaystyle \int \frac{1}{16 - (x-4)^2} \, dx$ using trigonometric substitution after completing the square.

When tackling integrals that involve rational functions or expressions that can be represented in terms of trigonometric identities, trigonometric substitution is a powerful technique. This type of substitution is especially useful when the integral contains expressions that can be likened to the forms found in the Pythagorean identity, such as a squared term minus another squared term. In this particular problem, the expression inside the integral, involving 16 minus the square of a binomial, suggests using a trigonometric substitution related to the identity 1 minus sine squared equals cosine squared, or similar identities involving tangent or cotangent functions.

Completing the square is an important step here. It simplifies the integrand into a form where standard trigonometric identities can be applied more easily. By rewriting the integrand, you transform the problem into a format where you can substitute x using sine, cosine, or tangent, thus simplifying the integral. This strategy not only makes it easier to evaluate the integral but also helps you understand the relationship between polynomials and trigonometric functions. Through this process, you not only solve the given integral but also practice converting algebraic expressions into trigonometric forms, which is a crucial skill in advanced calculus and analysis.

Using trigonometric substitution reinforces understanding of both integration techniques and trigonometric identities. It challenges you to consider how different mathematical concepts can be interconnected, and it provides a toolset for tackling a wide range of integral problems. This method is often applied in cases where the integral has boundaries defined by similar expressions, or whenever simplifying the expression into a more recognizable form makes integration feasible. As you continue practicing, you'll become more adept at choosing the right substitution for each unique integrand, ultimately enhancing your problem-solving skills in calculus.

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