At 06:33 AM 11/6/96 +0000, you wrote: >I posted a message recently (Integration...Constants) that received >several valuable responses requesting more information. Well, here is >the actual problem: > >In:=Integrate[(-2 x+ao)^-2 (-x+bo)^-1, x] > >Mathematica gives the following output: > >Out=-(1/((ao-2 bo)(ao- 2 x))) - (Log(-bo+x)/(ao^2 - 4 ao bo + 4 bo^2)) + > >Log(-ao + 2 x)/(ao^2 - 4 ao bo + 4 bo^2) > >This answer is incorrect. If performed by hand, the answer is shown to >have positive values for ao and bo. This is confirmed by the fact that >this integration is a classical result of chemistry. Years of chemical >reaction rate analysis show the values to be positive. Again, this >calculation was performed using the unix version of 2.2.3. Any help from >those who have already responded would be most appreciated.
I have done the following:
a. I put the problem in Mma 3.0 and I get basically the same thing that you received.
b. I asked myself, is this correct? Although I could have done the problem "by hand", I am lazy and I went to Handbook of Mathematical Tables and Formulas by Burington, 1948, (I used it in College) and found the integral and the solution (it is #84 on page 62). I then satisfied myself that your solution and my solution is the same as the solution in the Handbook. I will fax you the algebra, if you like.
c. From a mathematics point of view, the solutions above are incomplete. First the denominator term (ao-2*bo)^2 must be nonzero, or the solution is indeterminate. Second the argument of the Log must be non-negative, by definition of the Log, whether it is to the base e or some other base. With these two restrictions, the solutions should be complete. Incidentally, Mma or MathCad did not mention either of these restrictions, but Derive included the second one in its solution.
d. When you apply the integral to chemistry, one finds that one must add more restrictions. I believe in your case that ao and bo must be greater than zero or perhaps non-negative. The application of the integral is still valid, but with additional restrictions. In the case of Electrical Engineering, (I am an EE major), the appropriate integrals involve Resistance(R), Capacitance(C), Inductance(L), and frequency(f). From a mathematics viewpoint, these constants can be any value; but from an EE viewpoint, negative values of R,L,C or f are ludicrous and they must be restricted to non-negative values.
e. Generally speaking, I trust an integral when I can take the derivative of the integral and get the integrand. In some cases where I have taken a Taylor Series expansion and then integrated, the derivative differs from the integrand by a constant which can be explained.
Physics, Chemistry, Biology, and Engineering are applications of Mathematics where restrictions may be required.
Thanks for sharing the original problem. If I can help in the future, please call on me.