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A simple integration paradox?

Question

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

(1/2)(1/u)du

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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On 1/12/2018 at 11:53 AM, rocdocmac said:

1/(2x) = ½ln|x| + c

Spoiler

ln|2x| = ln|x| + ln(2), where ln(2) is a constant

Thus ...

"½ln|2x|" + c = ½(ln|x| + ln(2)) + c1 = ½ln|x| + c2 +c1 = ½ln|x| + c

 

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