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A simple integration paradox?



If the antiderivative of u^-1 = ln |u| + c then why does this not follow:

integrate 1/(2x) dx

set u =2x, then du = 2 dx, then dx = (1/2)du

Then we could integrate


By the definition above we get

1/2 ln|u| + c

which means that the integration of 1/(2x) = 1/2 ln|2x|| + c

However, this is a false statement.

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