Solving First Order ODE using the integrating factor approach 


I am trying to solve the differential equation, but I do not understand the method. Here is my working:
 

The step from line 3 to line 4 isn't correct: $\frac{di}{dt} e^t + ie^t = \frac{d}{dt}(ie^t)$, so line 4 should read $\frac{d}{dt}(ie^t) = 10t$ which can then be integrated directly.
In general, the ideas behind an integrating factor are outlined here as well as I could explain (if not better).
