\( \int \frac{d x}{x \sqrt{x^{2}-16}} \) ? (Hint: Scroll to either the header or the footer to see the formula page.) \( \frac{1}{4} \cos ^{-1}\left(\frac{4}{x}\right)+C \) \( \frac{1}{4} \sin ^{-1}\left(\frac{4}{x}\right)+C \) \( \frac{1}{4} \operatorname{sen}^{-1}\left(\frac{4}{x}\right)+C \) None of these responses is correct. Nest \( \left.\frac{4}{x}\right)+C \)

Let's take a little trip through the annals of calculus history! The technique used to evaluate integral expressions like \( \int \frac{d x}{x \sqrt{x^{2}-16}} \) is rooted in the method of trigonometric substitution. This method became prominent in the 17th century, with mathematicians like Isaac Barrow and later Leonhard Euler making significant contributions. They developed various approaches to simplify complex integrals, which laid the groundwork for modern integral calculus! Now, in practical terms, you'll often encounter this type of integral in physics and engineering, particularly when dealing with problems involving forces, center of mass, or in electrical engineering with circuit analysis. Understanding how to evaluate these integrals can be crucial for resolving real-world problems and helps in modeling various phenomena, such as the behavior of pendulums or electrical circuits with resistors. So keep honing those skills!