Re: Integrate[1/x, x] ??

• To: mathgroup at smc.vnet.net
• Subject: [mg29936] Re: [mg29919] Integrate[1/x, x] ??
• From: David Withoff <withoff at wolfram.com>
• Date: Thu, 19 Jul 2001 03:57:01 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```> Isn't the answer:
>
> ln(abs(x))+C
>
> and not log(x).
>
> My email address is anti-spammed.  Remove the 2 B's after hitting email reply
> if you want to email me.
>
> Tim 9-23

The absolute value function is not differentiable (the derivative is
different for different directions in the complex plane), so if x is
assumed in general to be complex, then the absolute value function
cannot be part of the anti-derivative of anything.  The natural logarithm
function in Mathematica is Log[x], and the derivative of Log[x] is 1/x
everywhere, so Log[x] is the correct antiderivative of 1/x.

The "+C" part of this is omitted for a variety of reasons.  Most tables
of integrals do not include the "+C" since it is clearly understood and
would just take up space.  The Integrate function is not intended to
return the most general possible anti-derivative anyway.  It is only
intended to return one antiderivative.  You can define your own function
to add the "+C" to every indefinite integral, but mostly it would just
be a nuisance and would take up space.

The result ln(abs(x))+C is the answer to something, such as if there
is an implied assumption that integration is along the real axis, but it
is not the answer to what Integrate is designed to do.

Dave Withoff
Wolfram Research

```

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