Cosine over Exponential - $\displaystyle \int_{-\pi/2}^{\pi/2} \frac{\cos x}{1+2^x}\ \mathrm{d}x$

$$ \begin{align*} I &= \int_{-\pi/2}^{\pi/2} \frac{\cos x}{1+2^x}\ \mathrm{d}x \\ &= \int_{-\pi/2}^{\pi/2} \frac{\cos (-x)}{1+2^{-x}}\ \mathrm{d}x\qquad \color{blue}{\text{[King's Rule]}}\\ &= \int_{-\pi/2}^{\pi/2} \frac{(\cos x)2^x}{1+2^{x}}\ \mathrm{d}x\\ I + I &= \int_{-\pi/2}^{\pi/2} \frac{(\cos x)(1+2^x)}{1+2^{x}}\ \mathrm{d}x = \int_{-\pi/2}^{\pi/2} \cos x\ \mathrm{d}x \\ 2I &= \sin x \Bigr|_{-\pi/2}^{\pi/2}\\ &= \sin(\pi/2) - \sin(-\pi/2)\\ &= 1 - (-1) = 2\\ I &= \boxed{1} \end{align*} $$