How can I find the limit of the integral?

  • #1
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

Homework Equations




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 can`t 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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Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,260
619

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

Homework Equations




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 can`t 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 :/

Have you thought about trying to apply l'Hopital's theorem?
 
  • #3
109
37
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
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)
 

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