Question

How do i solve the integral?

Answer 1

\[\int\limits_{0}^{+\infty}\frac{ x-2 }{ \sqrt{x}(x+1)^{k} }\]

Answer 2

How do i determine if it diverges or converges?

Answer 3

depends largely on \(k\) right?

Answer 4

yes

Answer 5

oh damn it is improper at both ends!!

Answer 6

?

Answer 7

it is an improper integral because you are going to infinity
that means in order for it to converge, the degree of the denominator has to be larger than the degree of the numerator by more than one

Answer 8

but you are also starting at \(0\) and the integrand is undefined there

Answer 9

there are other ways i think... you have to use the limit

Answer 10

for that to converge, the condition is reversed so i think there is no way to do this

Answer 11

you can solve it but i don't remember how xD

Answer 12

yes, i know it is a limit

Answer 13

I have the results in front of me now :D

Answer 14

but it is improper at both ends

Answer 15

but not the process, which I want to know ;)

Answer 16

what does the solution say?

Answer 17

the integral is asymptotic to
1/x^k

Answer 18

ok i got lost right there

Answer 19

m=1/2 (<1) and for +infty k>3/2

Answer 20

It says that the integral is asymptotic to 1/x^m

Answer 21

if we try \(k=1\) it diverges

Answer 22

i should shut up because i have no idea what "it is asymptotic to \(\frac{1}{x^m}\)" means
there is no \(m\) in it

Answer 23

I don't know either xD

Answer 24

Anybody?

Answer 25

damn that's a nasty one

Answer 26

Which Calc is this?

Answer 27

3?

Answer 28

I think it may be \(k > 3/2\)

Answer 29

But I'm not 100% sure. Let me confirm

Answer 30

Here's my thinking:
Let \(m=\) the degree of the numerator
Let \(n=\) the degree of the denominator
For it to converge, \(n - m > 1\)
Or, \((k + 1/2) - (1) > 1\)
\(k > 3/2\)

But I'm still not 100%