Determine whether this integral is convergent or divergent. If it is convergent, evaluate it. If it is diverge

Question

int (2, infinity) 1dx/(x+3)^(3/2)

Answer 1

It is convergent:
∫(2 to ∞) dx / (x+3)^(3/2)
= lim(a->∞) ∫(2 to a) dx / (x+3)^(3/2)
= lim(a->∞) [(x+3)^(-1/2) / (-1/2)] [2 to a]
= lim(a->∞) [-2(a+3)^(-1/2) + 2(5)^(-1/2)]
= 2/√5.

Answer 2

It is convergent as the place where the integrand, the expression inside the integral, goes to infinity is -3 which is outside the range of integration.

To evaluate it, put z = x+3, dz = dx and x going from 2 to infinity makes z go from 5 to infinity. The integral reduces to
int(5, infinity) z^(-3/2) dz, which is equal to 0 - 5^(-1/2)/(-1/2) .
this is 2/sqrt(5)

Answer 3

U = x + 3
dU = dx
U goes from 5 to infinity

int(5, infinity) U^-3/2

This is -2U^-1/2

As U goes to infinity -2(U^-1/2) goes to 0. The integral converges.

0 - -2/5^1/2 = 2/sqrt(5) = 0.894427191