Definite Integral Involving Trigonometric Substitution

Evaluate the definite integral from $4$ to $4\sqrt{3}$ of $\frac{1}{x^2 \sqrt{x^2 + 16}} \, dx$.

This problem involves evaluating a definite integral that can be tackled using the technique of trigonometric substitution. Trigonometric substitution is a method used in integration when the integrand contains expressions like the square root of a sum or difference of squares, such as $\sqrt{x^2 + a^2}$. This approach is particularly useful because it allows us to simplify the algebraic expression to a trigonometric identity, which can be more straightforward to integrate.

In this problem, the substitution can simplify the given integral into a form that's easier to work with. By making the right substitution, the square root expression can be transformed, and the integral simplifies into one involving a trigonometric function. The bounds of integration will also change according to the substitution, which needs careful handling when transforming back to the original variable.

Concepts from both integration techniques and trigonometric identities are essential here. Understanding the manipulation of integral bounds, recognizing suitable substitutions, and verifying results by differentiating are vital techniques in solving these kinds of integrals. You'll also see how this specific approach can be a stepping stone to solving more complex integrals that arise in calculus.

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