Chapter 8.3, Problem 2E

To determine

Evaluate the integral by the following steps.

a. Show that the substitution $x=\sqrt{2}\sec \theta$ transforms $I$ into $\frac{1}{2}\int \cos \theta d\theta$ and evaluate $I$ in terms of $\theta$.

b. Use a right triangle to show that with the above substitution $\sin \theta =\frac{\sqrt{x^2-2}}{x}$.

c. Evaluate $I$ in term of $x$.