(a) Evaluate: ∫_0^π / 2 e^x((1+ sin x)/(1+ cos x)) d x

$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sin x+\cos x}{\sqrt{\sin 2 x}} d x$

Put $\displaystyle \sin x-\cos x=t$ so that $\displaystyle (\cos x+\sin x) d x=d t$

$$=\int_{-\left(\frac{\sqrt{3}-$\displaystyle 1$}{$\displaystyle 2$}\right)}^{\frac{\sqrt{3}-$\displaystyle 1$}{$\displaystyle 2$}} \frac{d t}{\sqrt{$\displaystyle 1$-t^{$\displaystyle 2$}}}

$$$=\left[\sin ^{-1} t\right]_{-\left(\frac{\sqrt{3}-1}{2}\right)}^{\frac{\sqrt{3}-1}{2}}$

$$=$\displaystyle 2$ \sin ^{-$\displaystyle 1$}\left(\frac{\sqrt{3}-$\displaystyle 1$}{$\displaystyle 2$}\right)