rtoy
commented
2 weeks ago Imported from SourceForge on 2024-07-05 21:36:29 Created by aleksasd on 2011-09-07 10:11:15 Original: https://sourceforge.net/p/maxima/bugs/2270/#5995
WolframAlpha solution of this problem is not correct. The result is undefined for x=0 (in real domain).
Imported from SourceForge on 2024-07-05 21:36:28 Created by aleksasd on 2011-09-07 09:31:25 Original: https://sourceforge.net/p/maxima/bugs/2270
Wrong: (%i1) integrate(acos((x-1/2)/sqrt(1-x^2)),x); (%o1) -(%pi*x)/2
The result should be (%i2) sol:(2^(3/2)*x*acos((2*x-1)/(2*sqrt(1-x^2))) -sqrt(2)*asin((10*x+1)/(2*sqrt(7)*(x+1))) +sqrt(2)*asin((6*x-5)/(2*sqrt(7)*(x-1))) +asin((4*x-1)/sqrt(7)))/2^(3/2)$ Test of sol: (%i3) f:acos((x-1/2)/sqrt(1-x^2))$ (%i4) diff(sol,x)-f,radcan; (%o4) 0 (%i5) is(%=0); (%o5) true
Next a detailed solution has bug: (%i6) load(bypart)$ (%i7) assume(abs(x)<1); (%o7) [abs(x)<1] (%i8) byparts(f,x,f,x),factor$ (%i9) sol1:ev(%, nouns),factor$ Test of sol1: (%i10) diff(sol1,x)-f,radcan$ (%i11) is(%=0); (%o11) false
Note: this integral is from http://www.math.utexas.edu/pipermail/maxima/2011/025866.html (%i12) S1:'integrate(acos((2*c+k*u)/sqrt((1-k^2/4)*(1-u^2))),u)$ (%i13) subst([k=2/sqrt(5),c=-1/2/sqrt(5)],S1),factor; (%o13) integrate(acos((2*u-1)/(2*sqrt(1-u^2))),u)