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#