Hey guys, I have a question about improper integrals.

For example lets say the question is 1/(x-2)^3/2 dx and the interval is between infinity and 3.

How would I be able to determine if its convergent or divergent and if its convergent solve.

So I tried the problem out myself and what I did was I took the (x-2)^3/2 to the top so it becomes (x-2)^-3/2 and from there i just did the anti derivative of that which i thought was 1/(-3/2+1/1) giving me 1/(-1/4) which i then brought the 4 to the top, meaning the equation is now -4(x-2)^(-3/2+1/1) which turns out to be -4(x-2)^(-1/2) between the interval of infinity and 3. Is what i did so far correct or completely off? If it's wrong can you tell me why I can't do that and what the correct way of doing it is?

Sorry I had to make it long, i just wanted you guys to understand the problem I had.