Integral with product and limit

[festinalente]

Homework Statement

Lim (x→0) [ x^(-8) ∫ (0 to x^2) of (t-tan(t))/(1+t^2)dt ]

Homework Equations

The Attempt at a Solution

My first thought was to find the integral, then put (x^(-8)) into the solution of the integral and then do the limit.
I tried doing the substitution rule equating u= t-tan(t) and getting du=1-sec^2t dt. But then I had no idea of how to do the denominator as well as changing the upper and lower limits of the integral to correspond to the substitutions.
Any ideas on how to approach this question?
Thanks

 

[Dick]

I don't think you can 'do' the integral. I would try to expand (t-tan(t))/(1+t^2) in a power series around t=0 and integrate that. You don't need to keep very many terms. Why not?

 

[festinalente]

I don't think we're expected to know how to do a power series. I don't quite understand what you're suggesting to do. I looked at it again and tried to do two separate integrals
t/(1+t^2)dt - (tan(t))/(1+t^2)dt. It seems to be yielding no results at this point

 

[Dick]

I don't think we're expected to know how to do a power series. I don't quite understand what you're suggesting to do. I looked at it again and tried to do two separate integrals
t/(1+t^2)dt - (tan(t))/(1+t^2)dt. It seems to be yielding no results at this point

That approach is not going to yield results. There is no elementary integral for your expression. The power series is the easy way. Here's a suggestion for the harder way. Set u=x^8. Now you have lim u->0 of (1/u) times the integral from 0 to u^1/4 of (t-tan(t))/(1+t^2)dt, right? That's a difference quotient expression for the derivative with respect to u of the integral. Still with me? Use the Leibniz integral rule to do the derivative. Now take that expression, do another variable substitution (like u=v^(1/4)) and try to work out the limit as v->0 using l'Hopital. It's involved, but you have to work around not being able to do the integral somehow.

 

[Dick]

I don't think we're expected to know how to do a power series. I don't quite understand what you're suggesting to do. I looked at it again and tried to do two separate integrals
t/(1+t^2)dt - (tan(t))/(1+t^2)dt. It seems to be yielding no results at this point

Or a little simpler, maybe, just use l'Hoptial from the beginning. You don't really need the difference quotient thing.