How can I find the limit of the integral?

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1. Aug 9, 2017

SunGirl

1. The problem statement, all variables and given/known data
Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5

2. Relevant equations

3. The attempt at a solution
The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series first for the function sqrt(1 + 1/t^3) (but f(0) = infinity and I also cant use maclaurin series for sqrt(t^3+1) as t is infinity) and then take integral for the first several elements? please help, I don`t understand how should I solve this problem.
P.S. Sorry for my bad English :/

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2. Aug 9, 2017

Dick

Have you thought about trying to apply l'Hopital's theorem?

3. Aug 9, 2017

Gigaz

You need neither Maclaurin nor l'Hopital, this question is much simpler.
For very large x, the integral must be essentially the integral over t^(3/2) (Though you may want to find a solid reasoning for that, for example with a Taylor series). The rest is very simple.

4. Aug 9, 2017

SunGirl

Thank you very much!) Such a simple solution)