(a) Evaluate: ∫_0^π (e^cos x)/(e^cos x + e^- cos x) d x

Let $\displaystyle \mathrm{I}=\int_{0}^{\pi} \frac{\mathrm{e}^{\cos \mathrm{x}}}{\mathrm{e}^{\cos \mathrm{x}}+\mathrm{e}^{-\cos \mathrm{x}}} \mathrm{dx}$

$$\Rightarrow \mathrm{I}=\int_{$\displaystyle 0$}^{\pi} \frac{\mathrm{e}^{\cos (\pi-\mathrm{x})}}{\mathrm{e}^{\cos (\pi-\mathrm{x})}+\mathrm{e}^{-\cos (\pi-\mathrm{x})}} \mathrm{d} \mathrm{x}=\int_{$\displaystyle 0$}^{\pi} \frac{\mathrm{e}^{-\cos \mathrm{x}}}{\mathrm{e}^{-\cos \mathrm{x}}+\mathrm{e}^{\cos \mathrm{x}}} \mathrm{d} \mathrm{x}

$$Adding (i) and (ii), we get

$$\left.2 \mathrm{I}=\int_{$\displaystyle 0$}^{\pi} \mathrm{d} \mathrm{x}=\mathrm{x}\right]_{$\displaystyle 0$}^{\pi}=\pi, \quad \therefore \mathrm{I}=\frac{\pi}{2}