Evaluate the following integral using trigonometric substitution. \( \int \frac{d x}{\left(x^{2}-289\right)^{3 / 2}}, x>17 \) What substitution will be the most helpful for evaluating this integral? A. \( x=17 \sec \theta \) B. \( x=17 \tan \theta \) C. \( x=17 \sin \theta \)

The most helpful substitution for evaluating the integral \( \int \frac{d x}{\left(x^{2}-289\right)^{3 / 2}} \) is \( x=17 \sec \theta \). This is because the expression \( x^2 - 289 \) simplifies in terms of secant: Using this substitution, we have \( x^2 - 289 = 17^2 \sec^2 \theta - 289 = 289(\sec^2 \theta - 1) = 289 \tan^2 \theta \). Therefore, the integral then transforms into a trigonometric form that's simpler to manage, making the evaluation much more straightforward. After this substitution, the differential \( dx \) translates to \( dx = 17 \sec \theta \tan \theta d\theta \). When substituting this into the integral, you simplify the denominator significantly, avoiding complications and leading you directly to an integrable function. With the right transformation, you'll tackle the integral with ease!