One can consider the quantities

• $\int F_x\,dx=\int m\ddot{x}\,dx=\frac{1}{2}m(\dot{x}_f^2-\dot{x}_i^2)$
• The $y$ version of above
• The $z$ version of above

Are these what you're after? These three quantities aren't usually considered in standard problems, but they seem valid to me.

  Your more standard equation in 3D is the sum of the three bulleted equations here.

One can consider the quantities

• $\int F_x\,dx=\int m\ddot{x}\,dx=\frac{1}{2}m(\dot{x}_f^2-\dot{x}_i^2)$
• The $y$ version of above
• The $z$ version of above

Are these what you're after? These three quantities aren't usually considered in standard problems, but they seem valid to me. Your "Result 2" is the sum of the three bulleted equations here.