Answer
$\dfrac{7}{3}+\dfrac{e^2-e}{2}$
Work Step by Step
The work done is given by:
$W=\int_C F\cdot dr=\int_0^{1} (y^2+1)^2(2y dy)+ye^{y^2+1} dy=\int_0^{1} 2y (y^2+1)^2+ye^{y^2+1} dy$
Suppose $y^2+1=p \implies 2y dy =dp$
Thus, we have
$W=\int_C F\cdot dr=\int_1^2 p^2+\dfrac{e^p}{2} dp=[\dfrac{p^3}{3}+\dfrac{e^p}{2}]_1^{2}=[\dfrac{2^3}{3}+\dfrac{e^2}{2}]-[\dfrac{1^3}{3}+\dfrac{e^1}{2}]=\dfrac{7}{3}+\dfrac{e^2-e}{2}$
