Limits / Convergent integrals Question Watch
I just haven't even got a clue how to get started on this one
Thanks in advance.
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.