Let $f : R \rightarrow R$ be defined as $f ( x ) = e ^ { - x } \sin x$. If $F : [ 0,1 ] \rightarrow R$ is a differentiable function such that $F ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$, then the value of $\int _ { 0 } ^ { 1 } \left( F ^ { \prime } ( x ) + f ( x ) \right) e ^ { x } d x$ lies in the interval\\
(1) $\left[ \frac { 327 } { 360 } , \frac { 329 } { 360 } \right]$\\
(2) $\left[ \frac { 330 } { 360 } , \frac { 331 } { 360 } \right]$\\
(3) $\left[ \frac { 331 } { 360 } , \frac { 334 } { 360 } \right]$\\
(4) $\left[ \frac { 335 } { 360 } , \frac { 336 } { 360 } \right]$