Integral of 1 over the square root of x squared plus 16 using Trigonometric Substitution

Evaluate the integral $\displaystyle \int \frac{dx}{\sqrt{x^2 + 16}}$ using trigonometric substitution.

The problem of evaluating integrals such as $\int \frac{dx}{\sqrt{x^2 + 16}}$ often requires sophisticated techniques like trigonometric substitution. This method leverages the identities and properties of trigonometric functions to simplify the integrand into a form that is easier to integrate. Here, the presence of an expression of the form $\sqrt{x^2 + a^2}$ suggests that we can use a trigonometric identity to facilitate the integration process, specifically using the identity $x = a \tan(\theta)$ which transforms the integral into a trigonometric form more familiar and simpler to handle.

Trigonometric substitution is an advantageous strategy in integral calculus when encountering square roots involving quadratic expressions. This substitution method transforms the algebraic form into a trigonometric identity, simplifying the integration process. The key aspect of this technique is selecting the appropriate trigonometric function that matches the structure of $\sqrt{x^2 + a^2}, \sqrt{x^2 - a^2},$ or $\sqrt{a^2 - x^2}$. In this particular problem, the structure $\sqrt{x^2 + 16}$ suggests the use of tangent substitution, transitioning the problem into the realm of trigonometric integrals, which can often be processed using basic integration techniques once the substitution is applied.

Understanding when and how to apply trigonometric substitution not only makes specific problems more manageable but also deepens comprehension of integration techniques as a whole. By practicing these substitutions, students become more adept at recognizing patterns and selecting strategies that streamline the solution process, an essential skill set in tackling more complex integration problems. This problem exemplifies the effective use of trigonometric substitution as a powerful tool in integral calculus, reinforcing the intertwined nature of algebraic manipulation and trigonometric identities in mathematics.

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