Integral with product and limit

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Homework Help Overview

The problem involves evaluating the limit of a product of a function and an integral as x approaches 0, specifically focusing on the expression involving the integral of (t-tan(t))/(1+t^2) from 0 to x^2.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods for approaching the integral, including substitution and power series expansion. There is uncertainty about the feasibility of directly integrating the expression and whether a power series is required. Some participants attempt to separate the integral into two parts but find it unproductive.

Discussion Status

The discussion is ongoing, with participants exploring different strategies. Some suggest using power series while others express confusion about this approach. There is mention of using l'Hôpital's rule as a potential method, indicating a variety of perspectives on how to tackle the limit.

Contextual Notes

Participants note that there may be constraints on the expected knowledge, particularly regarding power series, which could affect the approaches taken. The complexity of the integral and the limit process is acknowledged, with some participants questioning the assumptions underlying their methods.

festinalente
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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
 
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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?
 
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
 
festinalente said:
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.
 
festinalente said:
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.
 

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