It looks like the given integral is
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy[/tex]
The integrand is continuous everywhere in the region D bounded by C, where
D = {(x, y) : 0 ≤ x ≤ 1 and x² ≤ y ≤ √x}
so by Green's theorem,
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = \iint_D \frac{\partial(10x+9\cos(y^2))}{\partial x} - \frac{\partial(5y+7e^x)}{\partial y} \, dx\, dy[/tex]
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = 5 \iint_D dx\, dy[/tex]
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = 5 \int_0^1 \int_{x^2}^{\sqrt x} dy\, dx[/tex]
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = 5 \int_0^1 (\sqrt x-x^2)\, dx[/tex]
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = 5 \left(\frac23 x^{\frac32} - \frac13x^3\right)\bigg|_0^1[/tex]
[tex]\displaystyle \int_C (5y+7e^x)\,dx + (10x+9\cos(y^2))\,dy = \boxed{\frac53}[/tex]
