f(x) = 1/e^x + e^x = e^-x + e^x

Int(e^x) = e^x + C

Using the chain rule, Int(e^-x) = Int[-(e^-x)/(-1)] = e^-x/(-1) + C

so, Int(f(x)) = -e^-x + e^x = e^x - e^-x + C

and from x = 0 to 1 Int(f(x) = (e^1 - e^-1) - (e^0 - e^-0)

= (e - 1/e) - (1 - 1)

**= e - 1/e**