Simplifying the Integral with Trigonometric Substitution

Simplify the integral $\displaystyle \, \int \frac{1}{\sqrt{16 - x^2}} \, dx$ using the substitution $x = 4 \sin \theta$.

In this problem, we employ trigonometric substitution to simplify the integral of 1 over the square root of 16 minus x squared. This type of problem showcases the usefulness of trigonometric identities and substitutions in integral calculus. Specifically, the substitution x equals 4 sin theta helps transform the integral into a more manageable form.

The substitution takes advantage of the Pythagorean identity, which is essential for this problem. By letting x equal 4 sin theta, we directly invoke the identity 1 minus sin squared theta equals cosine squared theta. This allows us to simplify the square root expression and rewrite the integral in terms of theta, where the integration process becomes straightforward.

Understanding when and how to apply trigonometric substitutions can be crucial for solving integrals involving square roots of quadratic expressions. This method is not only applicable to this particular problem but is also a strategic approach used in various complex integration problems, helping to break down seemingly challenging integrals into simpler trigonometric forms.

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