Sorry guy, idk how to do latex on here.
Here's my ATTEMPT.
If im correct, here's what I've learned from this exercise:
Can we simply ignore the jump at x=0?
No, course not. Look at the pic. It only became negligible because on both sides, g'(x) was CONSTANT and EQUAL on both sides.
What if it was constant but not equal? What if g(x) = -x+1 on x>0? That means g'(x) = -1 for x>0, which would change the answer to -F(1)-F(-1)+2F(0) = -e-1/e+2.
What if g'(x) wasn't constant? What if g'(x) = -sin(1/x), meaning it oscillates and the limit doesn't exist as x approaches 0 (on either side)? That would mean you would have to find the antiderivative of -sin(1/x)*e^x. I dunno what the answer to this question would be then.
What if g(x) is continuous everywhere but g'(x) exist nowhere? I dont wanna think how that would go...
If g'(x) was a nonconstant, but nice function, you'd still have to find the antiderivative of g'(x)e^x, which may or may not be doable.