# Limits / Convergent integrals QuestionWatch

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Thread starter 13 years ago
#1
Sorry about this but i've got another question I'd love some help with

I just haven't even got a clue how to get started on this one

Thanks in advance.

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13 years ago
#2
How rigorous an answer do you want?

The crucial facts are that:
x^a is integrable on (0,1) for a > -1
x^b is integral on (1,infinity) for b < -1.

Near 0 your integrable looks like

(x - x^3/6 - x + x^3/2)^2/x^q = 1/9 (x^6/x^q)

So you need 6-q > -1 i.e q < 7

For large x then the main term is x^(2-q) cos^2x which is integrable when 2-q < -1 i.e. q>3.

The integral will not be convergent when q = 7 as then the graph is akin to 1/(9x) near 0.
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Thread starter 13 years ago
#3
Thanks very much Rich, rep has been naturally given
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