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Topic: are the commercial results continuous?
Replies: 59   Last Post: Jul 30, 2012 4:28 PM

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 Nasser Abbasi Posts: 6,663 Registered: 2/7/05
Re: are the commercial results continuous?
Posted: Jul 17, 2012 1:21 AM

On 7/16/2012 4:11 PM, Nasser M. Abbasi wrote:
> On 7/16/2012 3:50 PM, clicliclic@freenet.de wrote:
>

>>
>
> Here they are, using V 8.04 on windows:
>
> ---- first one -----------------
> Clear[x];
> integrand=(2+3*x-2*x^2)/((1+x+x^2)^2*Sqrt[1-x+x^2]);
> sol=Integrate[integrand,{x,-1,0}]
> In[3]:= N[sol]
> Out[3]= -1.60465+0. I
>
> In[5]:= sol2=NIntegrate[integrand,{x,-1,0},WorkingPrecision->20]
> Out[5]= 0.061431946878726698474
> -----------------------
>
> --- second example ------
> integrand=(1-x+3*x^2)/((1+x+x^2)^2*Sqrt[1-x+x^2]);
> sol=Integrate[integrand,{x,-1,0}]
>
> In[8]:= N[sol]
> Out[8]= -1.84016-1.4803*10^-16 I
>
> In[9]:= sol2=NIntegrate[integrand,{x,-1,0},WorkingPrecision->20]
> Out[9]= 2.6027195829817573677
> -----------------------

I upgraded to Maple 16, and tried these on it. Maple seems to handle
these type of integrals better. Not only the expressions came out
smaller in size, but evaluating the symbolic result numerically
using evalf(sol) produced the correct result without having to do
numerical integration with increased precision.

--- example 1 Maple 16-----
integrand:=(2+3*x-2*x^2)/((1+x+x^2)^2*sqrt(1-x+x^2));
sol:=int(integrand,x=-1..0);

-3/2+(1/16)*sqrt(2)*sqrt(3)*ln(3*sqrt(2)+4)-(1/16)*sqrt(2)*sqrt(3)*
ln(3*sqrt(2)-4)+(1/2)*sqrt(3)-(1/16)*sqrt(2)*sqrt(3)*ln(sqrt(2)*
sqrt(3)+2)+(1/16)*sqrt(2)*sqrt(3)*ln(sqrt(2)*sqrt(3)-2)+(3/8)*
sqrt(2)*arctan(sqrt(2))

evalf(sol);
0.0614319480
--------------------------

---- example 2 Maple 16------------
integrand:=(1-x+3*x^2)/((1+x+x^2)^2*sqrt(1-x+x^2));
sol:=int(integrand,x=-1..0);

1+(1/12)*sqrt(2)*sqrt(3)*ln(3*sqrt(2)+4)-(1/12)*sqrt(2)*sqrt(3)*
ln(3*sqrt(2)-4)-(1/12)*sqrt(2)*sqrt(3)*ln(sqrt(2)*
sqrt(3)+2)+(1/12)*sqrt(2)*sqrt(3)*ln(sqrt(2)*sqrt(3)-2)+sqrt(2)*arctan(sqrt(2))

evalf(sol);
2.602719583
---------------------------------

I am glad to also say that the copy using the mouse from the
Maple worksheet seems to be working very well now in V 16. Now
I can copy things out like I did above with no problem. Before,
this did not work well at all.

--Nasser