00:01
This problem we are to match improper integrals from these stresses that we have here, in which we can compare using the comparison test for integrals.
00:13
Now, the comparison test for integral states that if we have functions f and g that are continuous on the interval a to infinity, such that f of x is less than equal to g of x and x, and both f and g are positive functions.
00:36
For all x, that's greater than or equal to a.
00:38
Then the integral from a to infinity of f of x d x converges if the integral from a to infinity of g of x dx converges.
00:54
And the second one is that the integral from a to infinity of g of x dx diverges if the integral from a to infinity of g of x dx diverges if the integral from a to infinity of f of x, dx diverges.
01:13
So for our first improper integral, we have the integral from 1 to infinity of dx over x cubed plus 4.
01:25
So in here we want to let f of x equal to 1 over x cubed plus 4, and our g of x would be equal to 1 over x cubed.
01:41
Now since x cubed plus 4 is greater than x cubed, then 1 over x cubed plus 4 will be less than 1 over x cubed.
01:53
So we could match this one to choice c.
01:59
And to determine the convergence, we would use this first implicate.
02:05
Implication for comparison test because we found a function that is greater than our integrand than and also this integral here for dx over x cubed since that is convergent by the p series test since its power is 3 which is greater than 1 then by the comparison test the improperly.
02:36
Integral from 1 to infinity of dx over x q plus 4 is also convergent.
02:42
Now for the second improper integral we have integral from 1 to infinity of 2 plus e raised to negative x over x.
02:54
So our f of x here is equal to 2 plus e raise to negative x over x.
03:02
Now e raised negative x is greater than 0, so 2 plus e raise a negative x will be greater than just 2...