Final answer:
To evaluate the integral, we can use a trigonometric substitution. Substitute x = sec(u), simplify the integrand, and apply a u-substitution to find the final result.
Step-by-step explanation:
To evaluate the integral ∫₂⁸ dx / (x² - 1)^(3/2), we can use a trigonometric substitution. Let's substitute x = sec(u), which implies dx = sec(u)tan(u) du. Making this substitution, the integral becomes:
∫ ₂⁸ sec(u)tan(u) du / (sec²(u) - 1)^(3/2). Simplifying further:
∫ ₂⁸ sec(u)tan(u) du / tan²(u)^(3/2). Canceling the tan(u) terms, we get:
∫ ₂⁸ sec(u) du / tan(u).
Using the trigonometric identity sec²(u) - 1 = tan²(u), we can rewrite the integral as:
∫ ₂⁸ sec(u) du / √(sec²(u) - 1).
This integral can be evaluated using a simple u-substitution. After integrating, substitute u back in terms of x to get the final result.