Asked by alex
Determine whether the integral is convergent or divergent.If it is convergent, evaluate it.
from 0 to infinity
e^(-y^1/2)dy
from 0 to infinity
e^(-y^1/2)dy
Answers
Answered by
Steve
To evaluate
∫e^(-√x) dx
let
z^2 = x
2z dz = dx
and now you have
∫e^-z 2z dz
Now let
u = z
dv = e^-z dz
du = dz
v = -e^-z dz
∫u dv = uv - ∫v du, so
∫2z e^-z dz = -2ze^-z + 2∫e^-z dz
= -2ze^-z - 2e^-z
Now evaluate that at the limits and you get
∫[0,∞] e^(-√x) dx = 2
∫e^(-√x) dx
let
z^2 = x
2z dz = dx
and now you have
∫e^-z 2z dz
Now let
u = z
dv = e^-z dz
du = dz
v = -e^-z dz
∫u dv = uv - ∫v du, so
∫2z e^-z dz = -2ze^-z + 2∫e^-z dz
= -2ze^-z - 2e^-z
Now evaluate that at the limits and you get
∫[0,∞] e^(-√x) dx = 2