Find a Function F Such That $\mathbf { F } = \nabla f$

Essay

Find a function f such that $\mathbf { F } = \nabla f$ and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the curve C. $\mathbf { F } ( x , y ) = e ^ { 2 y } \mathbf { i } + \left( 1 + 2 x e ^ { 2 y } \right) \mathbf { j } ; C : \mathbf { r } ( t ) = t e ^ { t } \mathbf { i } + ( 1 + t ) \mathbf { j } ; 0 \leq t \leq 1$