[tex]\mathbf f(x,y)=7xy^2\,\mathbf i+7x^2y\,\mathbf j[/tex]
[tex]\dfrac{\partial f}{\partial x}=7xy^2\implies f(x,y)=\dfrac72x^2y^2+g(y)[/tex]
[tex]\dfrac{\partial f}{\partial y}=7x^2y=7x^2y+\dfrac{\mathrm dg}{\mathrm dy}[/tex]
[tex]\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C[/tex]
[tex]\implies f(x,y)=\dfrac72x^2y^2+C[/tex]
By the gradient theorem, the line integral over the given path [tex]\mathcal C[/tex] parameterized by the given vector function [tex]\mathbf r(t)[/tex] is
[tex]\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{\mathcal C}\nabla f\cdot\mathrm d\mathbf r=f(\mathbf r(1))-f(\mathbf r(0))[/tex]
[tex]\implies\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=14-0=14[/tex]
