My best interpretation of the math here is that you're talking about the line integral,
[tex]\displaystyle \int_C \left(y+e^{\sqrt x}\right) \, dx + \left(2x + \cos\left(y^2\right)\right) \, dy[/tex]
I won't bother trying to decipher what look like multiple choice solutions.
By Green's theorem, the line integral above is equivalent to
[tex]\displaystyle \iint_D \frac{\partial\left(2x+\cos\left(y^2\right)\right)}{\partial x} - \frac{\partial\left(y+e^{\sqrt x}\right)}{\partial y} \, dx \, dy[/tex]
where D is the set
[tex]D = \left\{ (x, y) : 0 \le x \le 1 \text{ and } 0 \le y \le x^2 \right\}[/tex]
Compute the double integral:
[tex]\displaystyle \int_0^1 \int_0^{x^2} \left(2 - 1\right) \, dy \, dx = \int_0^1 \int_0^{x^2} dy \, dx = \int_0^1 x^2 \, dx = \boxed{\frac13}[/tex]
