Re: Simple Integrations in Mathematica

*To*: MATHGROUP at yoda.physics.unc.edu*Subject*: Re: Simple Integrations in Mathematica*From*: Jeffrey P. Golden <jpg at newton.macsyma.com>*Date*: Mon, 29 Mar 1993 20:04-0500

Date: Wed, 24 Mar 1993 10:44 EST From: winkel at nextwork.rose-hulman.edu 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] x 2 Sin[-] 2 Out[1]= --------------- x x Cos[-] - Sin[-] 2 2 In[2]:= Integrate[(1+Sin[x])/Cos[x]^2,{x,0,Pi/4}] 5/4 (1 + I) (1 + (-1) ) Out[2]= --------------------- 1/4 -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 macsyma.com> Organization: Macsyma Inc.