Answer
$\dfrac{-44}{15}$
Work Step by Step
The tangential form for Green Theorem is given as:
Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y}) dx dy $ ...(1)
As per the question, we have $M=(y+e^x \ln y) ; N=\dfrac{e^x}{y}$
Equation (1) becomes Counterclockwise Circulation: $\oint_C F \cdot T ds= \iint_{R} (\dfrac{\partial (\dfrac{e^x}{y})}{\partial x}-\dfrac{\partial (y+e^x \ln y)}{\partial y}) dx dy=\iint_{R} (\dfrac{e^x}{y}-\dfrac{e^x}{y}-\dfrac{e^x}{y}) dx dy$
This implies that $-\int_{-1}^{1} \int_{x^4+1}^{3-x^2} dy dx=-\int_{-1}^{1} [y]_{x^4+1}^{3-x^2}$
or, $-\int_{-1}^{1} (2-x^2-x^4) dx=-[2x-\dfrac{x^3}{3}-\dfrac{x^5}{5}]_{-1}^{1}$
Thus, $-4+\dfrac{2}{3}+\dfrac{2}{5}=\dfrac{-44}{15}$
