How can I tackle this problem with as few abstract ideas as possible when attempting to 'purely' integrate from left to right? I've tried a few different versions of this term, but I'm not sure how to get into an answer. Is there another way to approach this?
Just a bit of manipulation and you have the answer. Split the integral this way. $\int(t+1-\frac{1}{t}).e^{t+ \frac{1}{t}}+ \int(1-\frac{1}{t^2}).e^{t+ \frac{1}{t}}$ Now right side easily evaluates to $e^{t+ \frac{1}{t}}$. For the left the side multiply and divide it by $t$ and write it as $t(\frac{1}{t} + 1- \frac{1}{t^2})$ . Then the left integral becoems $\int e^{t+ \frac{1}{t}}dt + \int t(1-\frac {1}{t^2})e^{t+ \frac{1}{t}}dt$. Now here in the right side integral take $t$ as the 1st and the rest as the 2nd function(and using by parts integration).So the right integral on integrating becomes $t.e^{t+ \frac{1}{t}}$$- \int e^{t+ \frac{1}{t}}.dt$ which cancels the left most term of the integral hence we finally obtain the integral as $(t+1).e^{t+ \frac{1}{t}} + C$