I am reading Chapter 8 (Ramsey Model) of Introduction to Modern Economic Growth by Daron Acemoglu.
At section 8.1 (page 377), the text introduces a boundary condition for infinite-horizon case, known as No-Ponzi condition. It provides the outline of derivation to illustrate that the boundary condition is necessary, but I am not sure about the intermediate steps (which I appended below).
May I ask how to derive the RHS of equation from the LHS of equation please?
$$\int_0^T c(t) * e^{nt} * e^{-\int_0^t r(s) ds} dt = \int_0^T c(t) * e^{-\int_0^t (r(s) - n) ds} dt$$
$$ \int_0^T c(t) e^{nt} e^{-\int_0^t r(s) ds} \; dt = \int_0^T c(t) e^{-\int_0^t (r(s) - n) ds} \;dt $$ using $e^a e^b = e^{a+b}$ $$ \int_0^T c(t) e^{nt-\int_0^t r(s) ds} \; dt = \int_0^T c(t) e^{-\int_0^t (r(s) - n) ds} \;dt $$ $$ \int_0^T c(t) e^{\int_0^t n \; ds-\int_0^t r(s) ds} \; dt = \int_0^T c(t) e^{-\int_0^t (r(s) - n) ds} \;dt $$ $$ \int_0^T c(t) e^{-\int_0^t -n \; ds-\int_0^t r(s) ds} \; dt = \int_0^T c(t) e^{-\int_0^t (r(s) - n) ds} \;dt $$ because integration is a additive operation, for well behaved integrands we can merge the integrals $$ \int_0^T c(t) e^{-\int_0^t (r(s) -n) \; ds} \; dt = \int_0^T c(t) e^{-\int_0^t (r(s) - n) ds} \;dt $$