Need some help with improper Integrals

Question

Hello, I just need a little help finding the antiderative of this problem

Answer 1

When approaching this problem don't worry about the upper infinity limit to start.

1. Use u substitution to integrate e^-y/2, where dy = -2 du
2. The upper limit of +inf makes e^-y/2 go to zero.
3. Finish problem as usual and simplify: -2(33)[0-e^-4/2]

Answer 2

First make a change of variable by setting u = -y/2 ===> du/dy = -1/2 ==> dy = -2du

Your integral becomes

33∫₄^∞(e^-y/2)dy = 33∫₄^∞e^u(-2du) = -66∫₄^∞e^udu = -66[e^u]₄^∞ = -66[e^(-∞) - e^-2] = 66e^-2 ≅ 8.93

Answer 3

∫_{(from 4 to ∞)} e^-y/2dy = lim_b→∞ ∫_{(from 4 to b)} e^-y/2dy = lim_b→∞ [-2e^-y/2_{(from 4 to b)}] =

-2lim_b→∞ [e^-b/2 - e^-2] = -2(0 - e^-2) = 2 / e²