How do you determine if the improper integral converges or diverges $int dx/((3x-2)^6)$ from 2 to infinity?

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Answer 1 · 2017-01-24T17:10:07

The integral is convergent and:

$int_2^oo (dx)/(3x-2)^6 = 1/15360$

By definition:

$int_2^oo (dx)/(3x-2)^6 = lim_(t->oo) int_2^t (dx)/(3x-2)^6 = lim_(t->oo) -1/15 [1/(3x-2)^5]_2^t$

We have that:

$lim_(t->oo) 1/(3t-2)^5 = 0$

So:

$int_2^oo (dx)/(3x-2)^6 = 1/15 1/4^5 = 1/15360$

How do you determine if the improper integral converges or diverges int dx/((3x-2)^6) int dx/((3x-2)^6) from 2 to infinity?