How do you show whether the improper integral #int (79 x^2/(9 + x^6)) dx# converges or diverges from negative infinity to infinity?

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Answer 1 · 2016-03-18T13:25:24

I would integrate by trigonometric substitution, then check that the limit exists.

We can take out a constant factor, so #int_-oo^oo (79x^2/(9 + x^6)) dx# converges if and only if #int_-oo^oo (x^2/(9 + x^6)) dx# converges.

#int (x^2/(9 + x^6)) dx = 1/9tan^-1(x^3/3)#

As #xrarroo#, we have #tan^-1(x^3/3) rarr pi/2# (and as #xrarr-oo#, we have #tan^-1(x^3/3) rarr -pi/2#) so both

#int_-oo^0 (x^2/(9 + x^6)) dx# and #int_0^oo (x^2/(9 + x^6)) dx# converge.

Therefore, #int_-oo^oo (x^2/(9 + x^6)) dx# converges.