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Re: Simple Integrations in Mathematica

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  • Subject: Re: Simple Integrations in Mathematica
  • From: Jeffrey P. Golden <jpg at>
  • Date: Mon, 29 Mar 1993 20:04-0500

    Date: Wed, 24 Mar 1993 10:44 EST
    From: winkel at

    Concerning Michael Ibrahim's question of making Mathematica output  
    look like the textbook solutions for trig integrals, I say, "Let us  
    not worry about that!"  


Shouldn't one consider the specifics of Michael's question before saying 
that?  If Mathematica may be unnecessarily giving especially foolish 
answers from a mathematical point of view, shouldn't that be addressed?

Here, repeated, are Michael's examples:

In[1]:= Integrate[(1+Sin[x])/Cos[x]^2,x]

           2 Sin[-]
Out[1]= ---------------
            x        x
        Cos[-] - Sin[-]
            2        2

In[2]:= Integrate[(1+Sin[x])/Cos[x]^2,{x,0,Pi/4}]

        (1 + I) (1 + (-1)   )
Out[2]= ---------------------
            -I + (-1)

I find it troublesome that Mathematica is introducing half angles for 
a problem that can easily be done without them.  Returning answers 
using the same types of functions as in the input makes the answers 
more easily comprehensible, and easier to confirm by differentiation 
and simplification.  E.g. Macsyma returns  tan(x) + 1/cos(x)  for the 
problem in In[1].

But I find more surprising that you can easily accept the answer in 
Out[2] as a substitute for the easily obtained  Sqrt[2] !  The answer 
returned looks incomprehensible to me, unnecessarily involves complex 
quantities, and isn't even simplified ( (-1)^(5/4)  is just  -(-1)^(1/4) , 
not to mention  -Sqrt[I] .)

    When the zero was introduced and the Roman  
    numeral system was challenged, I doubt if the philosophy of those  
    wanting to move ahead with a more capable system of counting always  
    tried to get the new number writing system to look like Roman system.   
    We need to move on.  

What's wrong with examining whether the computer algebra systems we 
use are doing good, correct, comprehensible symbolic mathematics?

From: Jeffrey P. Golden <jpg at>
Organization: Macsyma Inc.

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