Evaluation of definite integral using contour integration

Question

It seems simple to evaluate $$\int_0^\pi \dfrac{d\theta}{5+4\cos \theta}$$ I tried with $$\int_0^\pi \dfrac{d\theta}{5+4\cos \theta}=\dfrac{1}{2}\int_0^{2\pi} \dfrac{d\theta}{5+4\cos \theta}=\dfrac{1}{2i}\oint_c \dfrac{dz}{(z+2)(2z+1)} $$ (c is a unit circle centered at the origin) which evaluates to $2\pi/3$. The answer given is $\pi/3$? I would like to know where did I go wrong.

Answer 1

$$z=e^{i\theta}\implies 2\cos\theta=z+1/z$$

$$\dfrac{1}{2i}\oint_c \dfrac{dz}{(z+2)(2z+1)}=\dfrac{2\pi i}{2i}\lim_{z\rightarrow-1/2}\dfrac{z+1/2}{(z+2)(2z+1)}=\frac{\pi}{3}$$

Using cauchy residue formula.

Answer 2

$$R(-2) = 0 \text{ as it is outside} |z| = 1$$

$$R(\frac{-1}{2}) = \lim_{z\to-1/2}\frac{z+1/2}{(z+2)(2)(z+1/2) } = \frac{1}{2(3/2)} = \frac{1}{3}$$

So, $$I = \frac{1}{2i}\oint_{|z|=1}\frac{dz}{(z+2)(2z+1) } = \frac{1}{2i} \times2\pi i \times\text{ sum of residues } = {\pi}\times\frac{1}{3} = \frac{\pi}{3}$$