Answer
$\frac{\pi}{2}-\frac{1}{6}\times\sqrt 2 $
Work Step by Step
$\int_{c} z dx+xy dy+y^{2}dz$
Line integral where $c$ is the curve: $x=\sin t, z= \tan t, y= \cos t$
$\displaystyle\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan t \times \cos t +\sin^{2}t\times \cos t\times(-\sin t) +\cos^{2}t\times \sec^{2}t$
=$\left[-\cos t-\frac{1}{3}\sin^{3}t+t\right ] | ^{\frac{\pi}{4}}_{\frac{-\pi}{4}}$
= $[(-\frac{1}{\sqrt 2}-\frac{1}{3}\times (\frac{1}{\sqrt 2})^{3}+\frac{\pi}{4})-(-(\frac{1}{\sqrt 2})-(\frac{1}{3}\times (\frac{1}{\sqrt 2})^{3})-\frac{\pi}{4})] $
= $\frac{\pi}{4}+\frac{\pi}{4}-\frac{1}{3}\times\sqrt 2$
= $\frac{\pi}{2}-\frac{1}{6}\times\sqrt 2 $
