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so i was messing around and found that

is always true.

taking the integrand as x

at first we have

but the limit is 2 sided(as m approaches the positive side of zero i get a positive value, while as m approaches the negative side of zero, i get a negative value.)

So i decided to find the average of both sides of the limit:

i graphed this with m=a really small (positive) number and found that it converges almost exactly to ln(x)

anyway, i just thought it was pretty cool how the power property always holds so i figured id share it. most of the sites i see just say its ln(x), period.

is always true.

taking the integrand as x

^{-1}at first we have

but the limit is 2 sided(as m approaches the positive side of zero i get a positive value, while as m approaches the negative side of zero, i get a negative value.)

So i decided to find the average of both sides of the limit:

i graphed this with m=a really small (positive) number and found that it converges almost exactly to ln(x)

anyway, i just thought it was pretty cool how the power property always holds so i figured id share it. most of the sites i see just say its ln(x), period.

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