Chapter 8.3, Problem 4E

To determine

Evaluate the integral by the following steps.

a. Show that the substitution $x=2\tan \theta$ transforms $I$ into $\frac{1}{8}\int {\cos }^2\theta d\theta$.

b. Use the formula $\int {\cos }^2\theta d\theta =\frac{1}{2}\theta +\frac{1}{2}\sin \theta \cos \theta$ to evaluate $I$ in terms of $\theta$.

c. Show that $\sin \theta =\frac{x}{\sqrt{x^2+4}}\text{and}\cos \theta =\frac{2}{\sqrt{x^2+4}}$.

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