$$\left. \begin{array} { l } { M = ( x + 2 ) \operatorname { sin } ( y ) \quad N = x \operatorname { cos } ( y ) } \\ { M _ { y } = ( x + 2 ) \operatorname { cos } ( y ) \quad N x = \operatorname { cos } y } \\ { \because M y \neq N x \quad \therefore \text { not exact } } \\ { x e ^ { x } ( x + 2 ) \operatorname { sin } ( y ) + x ^ { 2 } e ^ { x } \operatorname { cos } ( y ) y }\\{\therefore \text{now the function is exact}} \end{array} \right.$$
$$\left. \begin{array} { l }{\exists~\varphi(x,y)~s.t.\quad\varphi = \int M d x }\\{ = \int x e ^ { x } ( x + 2 ) \operatorname { sin } ( y ) d x }\\{ = \int ( x ^ { 2 } e ^ { x } + 2 x e ^ { x } ) \operatorname { sin } ( y ) }\\{ = ( x ^ { 2 } e ^ { x } ) \operatorname { sin } ( y ) + h ( y ) }\\{\varphi_y = ( x ^ { 2 } e ^ { x } ) \operatorname { cos } ( y ) + h ^ { \prime } ( y ) + h ^ { \prime } ( y ) + h ^ { \prime } ( y ) }\\{h^{\prime}(y)=0}\\{h(y)=\text{constant}}\\{\varphi=(x^2e^x)\sin y=C.}\end{array} \right.$$
