Consider the integral \(\int_{0}^{2 \pi} \frac{d \theta}{5-4 \cos \theta}\).

a. Evaluate this integral by making the substitution \(2 \cos \theta=z+\frac{1}{z}\), \(z=e^{i \theta}\), and using complex integration methods.

b. In the 1800 , Weierstrass introduced a method for computing integrals involving rational functions of sine and cosine. One makes the substitution \(t=\tan \frac{\theta}{2}\) and converts the integrand into a rational function of \(t\). Note that the integration around the unit circle corresponds to \(t \in(-\infty, \infty)\).

i. Show that

\[\sin \theta=\frac{2 t}{1+t^{2}}, \quad \cos \theta=\frac{1-t^{2}}{1+t^{2}}\]

ii. Show that

\[d \theta=\frac{2 d t}{1+t^{2}}\]

iii. Use the Weierstrass substitution to compute the above integral.