#2156 integrate_sqrt_of_trig

[d-stanislav]

(%i1) integrate(sqrt(1-cos(x)^2),x,0,%pi/2);
(%o1) - 1 ( right answer: 1 )
(%i2) integrate(sqrt(1-sin(x)^2),x,0,%pi/2);
Is tan(x) positive or negative?p;
(%o2) - 1 ( right answer: 1 )
-------------------------------------------------------------
Maxima version: 5.23.2
Maxima build date: 17:9 1/17/2011
Host type: i686-pc-mingw32
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.8
-------------------------------------------------------------

[Dieter Kaiser]

Both examples are correct, when setting the option variable radexpand to false:

(%i1) radexpand:false$

(%i2) integrate(sqrt(1-cos(x)^2),x,0,%pi/2);
(%o2) 1

(%i3) integrate(sqrt(1-sin(x)^2),x,0,%pi/2);
(%o3) 1

By the way, the indefinite integrals are more correct too:

(%i4) integrate(sqrt(1-cos(x)^2),x);
(%o4) -1/sqrt(tan(x)^2+1)

(%i5) integrate(sqrt(1-sin(x)^2),x);
(%o5) tan(x)/sqrt(tan(x)^2+1)

I think it is a general problem, that the option variable radexpand introduces wrong simplifications.

Dieter Kaiser

[d-stanislav]

Unfortunately, option radexpand:false does not give a right results:
(%i1) radexpand:false;
(%o1) false(%i2) integrate(sqrt(1-cos(x)^2),x,0,%pi);
(%o2) 0 (wrong)
(%i3) integrate(abs(sin(x)),x,0,%pi);
(%o3) 2 (right, but occasionally)
(%i4) integrate(sqrt(1-cos(x)^2),x,0,2*%pi);
(%o4) 0 (wrong)
(%i5) integrate(abs(sin(x)),x,0,2*%pi);
(%o5) 0 (wrong)
(%i6) wxplot2d(sqrt(1-cos(t)^2),[t,0,2*%pi],[x,-1,7],[y,-2,2]);
(%t6) << Graphics >>
(%i7) integrate(sqrt(1-sin(x)^2),x,0,%pi);
(%o7) 0 (wrong)
(%i8) integrate(abs(cos(x)),x,0,%pi);
(%o8) integrate(abs(cos(x)),x,0,%pi) (no action)

[sslavi]

I can confirm this bug on my system too. Actually, I am not even getting to the point where it asks "Is tan(x) positive or negative?". Please find enclosed the screenshot of all the options I have tried.

System info:
Maxima 5.36.1
SBCL 1.2.7
Win 8.1 64-bit

[sslavi]

The radexpand:false$ instruction doesn't help either. Actually, it increases the number of cases which end with an error message. See the screenshot.

Why is Maxima going using the Atanh() function anyways, in this case?