# How can I find the limit of the integral?

SunGirl

## Homework Statement

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

## 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 dont understand how should I solve this problem.
P.S. Sorry for my bad English :/

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Dick
Homework Helper

## Homework Statement

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

## 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 dont understand how should I solve this problem.
P.S. Sorry for my bad English :/

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

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.

SunGirl
SunGirl
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.
Thank you very much!) Such a simple solution)