Contour integration of cosine of a complex number I am trying to find the value of 
$$ -\frac{1}{\pi}\int_{-\pi/2}^{\pi/2} \cos\left(be^{i\theta}\right) \mathrm{d}\theta,$$
where $b$ is a real number.
Any helps will be appreciated!

Because cosine is an even function you may write the integral as
$$-\frac1{2 \pi} \int_{-\pi/2}^{3 \pi/2} d\theta \, \cos{\left ( b e^{i \theta} \right )} = -\frac{1}{i 2 \pi} \oint_{|z|=1} dz \frac{\cos{b z}}{z} $$
which, by the residue theorem or Cauchy's theorem, is 
$$-\frac{1}{i 2 \pi} i 2 \pi \cos{0} = -1$$
