Hi so I read online that the vector line integral
can be represented this way. But I also read that we can use 
when derivatives appear inside the integral.
So why is the last line still in terms of line integrals instead of definite integrals using integration by substitution?
The second to last line is in terms of definite integrals as you say. The working then proceeds to 'simplify' the result by writing it as a series of line integrals.