Which of the following is an improper integral?
a) integral from 0 to 3 of (x+1)/(3x-2) dx
b) integral from 1 to 3 of (x+1)/(3x-2) dx
c) integral from -1 to 0 of (x+1)/(3x-2) dx
d) None of these
Please help I don't know which one?
correct.
as someone else already observed,
(x+1)/(3x-2) goes to ∞ at x = 2/3
So, which interval contains this critical point?
answer A?
To determine which of the given integrals is improper, we need to examine if any of them meet the criteria for being classified as an improper integral.
An improper integral occurs when at least one of the following conditions is met:
1) The interval of integration is infinite
2) The function being integrated has an infinite discontinuity within the interval
3) The function being integrated has a vertical asymptote within the interval
Looking at the options:
a) The integral from 0 to 3 is a definite integral where both endpoints are finite (not infinite), and the function (x+1)/(3x-2) is continuous within the interval. Therefore, this is not an improper integral.
b) The integral from 1 to 3 is also a definite integral with finite endpoints, and the function (x+1)/(3x-2) is continuous within the given interval. So, this is not an improper integral either.
c) The integral from -1 to 0 is another definite integral with finite boundaries, and the function (x+1)/(3x-2) is continuous within the interval. Thus, this is not an improper integral.
Since none of the given options satisfy any of the criteria for being an improper integral, the correct answer is d) None of these.