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Re: integration

• To: mathgroup at smc.vnet.net
• Subject: [mg95970] Re: integration
• From: Bill Rowe <readnews at sbcglobal.net>
• Date: Sat, 31 Jan 2009 01:13:10 -0500 (EST)

On 1/30/09 at 5:47 AM, t.p.nixon at open.ac.uk wrote:

>Hi, I don't use mathematica but the other day I wanted to solve an
>integral and I'm out of practice so I went on-line to a mathematica
>based integrator.

>http://integrals.wolfram.com/index.jsp

>I typed in 1/sqrt(x^2-b^2) this should have an integral of
>cosh^-1(x/ b).  Ok, so it didn't recognize cosh but its answer was
>(I think) wrong.  it comes back with ln(x+sqrt(x^2-b^2)) it seems to
>have overlooked a quotient of b in the log.

>Is this a general problem with mathematica or just the on-line
>version or is it me?

It is you. Version 7 gives the following:

In[1]:= Integrate[1/Sqrt[x^2 - b^2], x]

Out[1]= Log[2*(Sqrt[x^2 - b^2] + x)]

In[2]:= D[%, x] // Simplify

Out[2]= 1/Sqrt[x^2 - b^2]

confirming the result returned is the anti-derivative of
1/Sqrt[x^2-b^2]. And since,

In[3]:= D[Log[a (Sqrt[x^2 - b^2] + x)], x] // Simplify

Out[3]= 1/Sqrt[x^2 - b^2]

It is clear the factor of two in the answer returned by Version
7 but not in the result from the on-line version is an arbitrary constant.

Further a quick check of tabulated integrals I have handy also
gives the same result as the on-line version for this integral