Problem on a question about the gravitation potential

[Garlic]

Hello everyone

1. Homework Statement

In the solution of a question, regarding the gravitational force fields and gravitational potential when the mass is inside Earth (where the height of the mass is lower than the radius of earth), the step where we take the integral of the force in order to get the potential is not very clearly defined, and I have problems understanding why it looks like this:

Where does this blue marked term come from? Why isn't its integral being taken, like the term on the right?
Why can't we simply find the potential by just taking the integral of the force field (without the blue marked term)?

(Where the G_N is the gravitation constant, R the radius of earth, m the mass of an object, and m_E the mass of earth, r cap the unit vector of the r coordinate)

Thank you

Homework Equations

The Attempt at a Solution

 

[PeroK]

To calculate the potential this way, you need to take the integral from $r = \infty$ to $r = r_m$. The equation for the force is different outside the Earth than within it, so you have to split the integral into two. The first integral is then from $r = \infty$ to $r = R$ and that's the first term, which is just the potential at the Earth's surface.

 

[Garlic]

Oh, now I understand it.. Thank you