$\int_{0}^{\frac{\pi }{2}}\frac{\left ( cosx+isinx \right )^{2}}{cos^{2}x+sin^{2}x}dx$
$=\int_{0}^{\frac{\pi }{2}}\left ( cos^{2}x-sin^{2}x+isin2x \right )dx$
$=\int_{0}^{\frac{\pi }{2}}\left ( cos2x+isin2x \right )dx$
$=\left [ \frac{sin2x}{2}-\frac{icos2x}{2} \right ]_{0}^{\frac{\pi }{2}}$
$= \frac{\sin \pi }{2}-i\frac{\cos \pi }{2}-\left ( 0-\frac{i}{2} \right )$
$=-\left ( \frac{-i}{2} \right )+\frac{i}{2}$
$=i$