>Doug Stewart wrote: >> root wrote: >>>>> My copy of Schaums (1968, printing 4) shows >>>>> >>>>> 14:334: >>>>> >>>>> int(1/(x*sqrt(x^n-a^n)),x) == 2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)) >>>>> >>>>> It seems this cannot be the answers. >>>>> Can someone with a later version please check for a typo? >>>>> >>>>> Tim >>>>> >>>>> >>>>> _______________________________________________ >>>>> Axiom-developer mailing list >>>>> Axiom-developer@nongnu.org >>>>> http://lists.nongnu.org/mailman/listinfo/axiom-developer >>>>> >>>>> >>>> My schaums shows that answer. >>>> also usind Maxima to do the derivative I get the LHS. >>>> (%i5) diff(2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)),x); >>>> (%o5) (a^n*x^(-n-1))/(sqrt(a^n)*sqrt(a^n/x^n)*sqrt(1-a^n/x^n)) >>>> (%i6) radcan(%); >>>> (%o6) 1/(x*sqrt(x^n-a^n)) >>>> >>> >>> If you compute >>> aa:=integrate(1/(x*sqrt(x^n-a^n)),x) >>> bb:=2/(n*sqrt(a^n))*acos(sqrt(a^n/x^n)) >>> cc1:=aa.1-bb >>> cc2:=aa.2-bb >>> >>> Can you find a simplification path (in Axiom) such that either cc1 or >>> cc2 simplify to a constant? >>> >>> Alternatively, can you use Maxima to find the constant? >>> >>> I'm failing to do either, although I'm still trying. >>> >>> Tim >>> >>> >> >> Maxima seems to give a wrong answer for this integration. >> >> >> (%i1) aa:integrate(1/(x*sqrt(x^n-a^n)),x); >> Is a positive or negative?p; >> >> (%o1) (2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n) >> >> and >> (%i2) aa:integrate(1/(x*sqrt(x^n-a^n)),x); >> Is a positive or negative?n; >> (%o2) >> log((2*sqrt(x^n-a^n)-2*sqrt(-a^n))/(2*sqrt(x^n-a^n)+2*sqrt(-a^n)))/(sqrt(-a^n)*n) >> >> >> >> >> I'm not good enough at simplifying to help you. >> Doug >> >> >> _______________________________________________ >with help from Maxima >> >Can you show me how to start with >> > >> >(2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n) >> > >> >and differentiate it to get >> > >> >1/(x*sqrt(x^n-a^n) >> > >> >using Maxima. >> > >(%i4) (2*atan(sqrt(x^n-a^n)/a^(n/2)))/(a^(n/2)*n)$ >(%i5) ratsimp(diff(%,x)); >(%o5) 1/(x*sqrt(x^n-a^n)) > >Barton > >so Maxima is not wrong.
None of these system are wrong and I believe that both expressions are equal up to a constant. What I'm looking for is a sequence of commands that will reduce the difference of (Axiom's result)-(Schaums result) to that constant. The subtle part is that there are many possible reduction paths and the order of the transformations matter. It seems pretty trivial to walk into a corner that will no longer reduce automatically. Simplification is one of the hardest problems. The problems in Schaums from 14.354 - 14.359 all seem to include the term: tan(%pi/4-(a*x)/2) which seems to be a source of difficulty for the simplifications. The original author had some insight that introduced this term and I can't figure out what transformation I can use to reverse it. The %pi/4 is clearly 45 degrees but I don't see many quarter-angle reductions anywhere. Tim _______________________________________________ Axiom-developer mailing list Axiom-developer@nongnu.org http://lists.nongnu.org/mailman/listinfo/axiom-developer
