1 | initial version |
Indeed, the integral of a function positive on $\left(0\ 2\pi\right)$-{$\pi$} can't be null.
The problem is with the primitive choosen by maxima
:
sage: integrate(sqrt(1-cos(x)),x)
-2*sqrt(2)/sqrt(sin(x)^2/(cos(x) + 1)^2 + 1)
i. e. $-\frac{2 \, \sqrt{2}}{\sqrt{\frac{\sin\left(x\right)^{2}}{{\left(\cos\left(x\right) + 1\right)}^{2}} + 1}}$.
This function brutally changes the sign of its derivative for $x=\pi$. This smell of an error in manipulating an absolute value.
Indeed, one can (more or less easily) show that :
sage: ((cos(2*x)==cos(2*x).trig_expand().subs(cos(x)^2==1-sin(x)^2))*-1)
....: .subs(x==x/2)+1
-cos(x) + 1 == 2*sin(1/2*x)^2
Therefore, we are trying to compute $\displaystyle\int_0^{2\pi}\sqrt{2\sin(x/2)}\,dx$, which is :
with assuming(x,"real"):integrate(sqrt(2*sin(x/2)^2).simplify(),x,0,2*pi,h
....: old=True)
integrate(sqrt(2)*abs(sin(1/2*x)), x, 0, 2*pi)
i. e. $\int_{0}^{2 \, \pi} \sqrt{2} {\left| \sin\left(\frac{1}{2} \, x\right) \right|}\,{d x}$.
since $x\in[=(0~2\pi)\Rightarrow\sin(x/2)>0$, this should be evaluated as
sage: integrate(sqrt(2)*sin(x/2),x,0,2*pi,hold=True)==integrate(sqrt(2)*sin(x/2)
....: ,x,0,2*pi)
integrate(sqrt(2)*sin(1/2*x), x, 0, 2*pi) == 2^(5/2)
But it's not.
What other possibilities do we have ? Both sympy
and giac
fail to
find a primitive of $\sqrt{1-\cos(x)}$ or the definite integral. But
fricas
does find a primitive :
sage: integrate(sqrt(1-cos(x)),x, algorithm="fricas")
-2*(cos(x) + 1)*sqrt(-cos(x) + 1)/sin(x)
i. e. $-\frac{2 \, {\left(\cos\left(x\right) + 1\right)} \sqrt{-\cos\left(x\right) + 1}}{\sin\left(x\right)}$.
which can be differentiated back to the original expression :
sage: integrate(sqrt(1-cos(x)),x, algorithm="fricas").diff(x).expand().trig_simp
....: lify()
sqrt(-cos(x) + 1)
But fricas
fails to find the definite integral, which we have to find
by hand :
sage: foo=integrate(sqrt(1-cos(x)),x, algorithm="fricas")
sage: (foo.limit(x=2*pi,dir="-")-foo.limit(x=0,dir="+")).simplify()
4*sqrt(2)
maple
finds a quite different expression :
sage: maple.integrate(sqrt(1-cos(x)),x).sage()
4*(cos(1/2*x)^2 - 1)*cos(1/2*x)/(sqrt(-2*cos(1/2*x)^2 + 2)*sin(1/2*x))
i. e. $\frac{4 \, {\left(\cos\left(\frac{1}{2} \, x\right)^{2} - 1\right)} \cos\left(\frac{1}{2} \, x\right)}{\sqrt{-2 \, \cos\left(\frac{1}{2} \, x\right)^{2} + 2} \sin\left(\frac{1}{2} \, x\right)}$
whch can been redifferentiated to (something close to) the form we have seen
when solving manually in order to understand
maxima
's error :
sage: maple.integrate(sqrt(1-cos(x)),x).sage().diff(x).trig_expand().factor().tr
....: ig_simplify()
sqrt(2)*sin(1/2*x)^2/sqrt(sin(1/2*x)^2)
i. e. $\frac{\sqrt{2} \sin\left(\frac{1}{2} \, x\right)^{2}}{\sqrt{\sin\left(\frac{1}{2} \, x\right)^{2}}}$.
Again, since $x\in\left(0~2\pi\right)$, this amounts to $\sqrt{2\sin(x/2)}$, which we used previously.
maple
fails to find the definite integral, which may be recomuted by
taking the limits at the bounds.
mathematica
gives yet another form :
sage: mathematica.Integrate(sqrt(1-cos(x)),x).sage()
-2*sqrt(-cos(x) + 1)*cot(1/2*x)
i. e. $-2 \, \sqrt{-\cos\left(x\right) + 1} \cot\left(\frac{1}{2} \, x\right)$.
which can also be redifferentiated to the same form as maple
's
result :
sage: mathematica.Integrate(sqrt(1-cos(x)),x).sage().subs(1-cos(x)==2*sin(x/2)^2
....: ).diff(x).trig_simplify()
sqrt(2)*sin(1/2*x)^2/sqrt(sin(1/2*x)^2)
i. e. $\frac{\sqrt{2} \sin\left(\frac{1}{2} \, x\right)^{2}}{\sqrt{\sin\left(\frac{1}{2} \, x\right)^{2}}}$.
mathematica
finds the definite integral :
sage: mathematica.Integrate(sqrt(1-cos(x)),[x,0,2*pi]).sage()
4*sqrt(2)
Not seen : the plots of the integrand and of the various solutions are also helpful to understand the problem, understand the possible errors of the various CASes and to check them against one another (hint : plot two finctions to be compared with a slight vertical shift alowing to separate them).
Numerical integration may also help...