Evaluate the given improper integral

Question

∫from 1 to 0 (e^(-1/x))/(x^2) PLEASE HELP ASAP. A VIDEO WOULD ALSO BE GREATLY APPRECIATED.

Answer 1

We see that the integrand of ∫_{(0 to 1)} e^-1/x/x²dx is undefined at x = 0.

So, ∫_{(0 to 1)}e^-1/x/x²dx = lim _b→₀+[ ∫_{(b to 1)} e^-1/x/x²dx].

Let u = -1/x. Then du = 1/x² dx.

∫e^-1/x/x²dx = ∫e^udu = e^u + C = e^-1/x + C

We have lim_b→0⁺[e^-1 - e^-1/b] = 1/e - e^-∞ = 1/e - 0 = 1/e.

Therefore, the integral converges to 1/e.

Answer 2

Use u substitution. Set u = 1/x which implies that x = 1/u, then dx = -1/u^2 du. (remember to change your bounds). Now the integral becomes from infinity to 1 of e^-u * u^2 * -1/u^2 = integral from 1 to infinity of e^-u. Evaluted it becomes e^-1 which is 1/e. That is the final answer. Hope this helps!!