Edit detail for Symbolic Integration revision 1 of 15

	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Editor: 127.0.0.1 Time: 2007/11/15 20:25:38 GMT-8
Note: transferred from axiom-developer

---

changed:
-
Errors in symbolic integration

  AXIOM Examples

  1)

    \begin{axiom}
    integrate(sin(x)+sqrt(1-x^3),x)
    \end{axiom}

    \begin{reduce}
    int(sin(x)+sqrt(1-x^3),x);
    \end{reduce}

  2)

    \begin{axiom}
    integrate(sqrt(1-log(sin(x)^2)),x)
    \end{axiom}

    \begin{reduce}
    int(sqrt(1-log(sin(x)^2)),x);
    \end{reduce}


  3)

    \begin{axiom}
    integrate(sqrt(sin(1/x)),x)
    \end{axiom}

    That seems strange given the claims about the "completeness" of
    Axiom's integration algorithm! But to be fair, Maple also returns
    this integral unevaluated.

    \begin{reduce}
    int(sqrt(sin(1/x)),x);
    \end{reduce}


  4)

    \begin{axiom}
    integrate(sqrt(sin(x)),x)
    \end{axiom}

    \begin{reduce}
    int(sqrt(sin(x)),x);
    \end{reduce}


    For this Maple 9 gives the following result:

    \begin{eqnarray}
    -{\frac {\sqrt {1+\sin \left( x \right) }\sqrt {-2\,\sin \left( x
    \right) +2}\sqrt {-\sin \left( x \right) }}{\cos \left( x \right) \sqrt {\sin \left( x
    \right) }}} \times
    \\
    \left( 2\,{\it EllipticE}
    \left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2} \right) -{\it 
    EllipticF} \left( \sqrt {1+\sin \left( x \right) },1/2\,\sqrt {2}
    \right)  \right)
    \end{eqnarray}

    And Mathematica 4 gives:

    \begin{equation}
    -2\,{\it EllipticE}(\frac{\frac{\pi }{2} - x}{2},2)
    \end{equation}

From unknown Tue Mar 22 11:48:00 -0600 2005
From: 
Date: Tue, 22 Mar 2005 11:48:00 -0600
Subject: symbolic integration
Message-ID: <20050322114800-0600@page.axiom-developer.org>

 \begin{axiom}
 integrate(exp(-x^2),x)
  \end{axiom}


From unknown Wed Mar 23 08:23:21 -0600 2005
From: 
Date: Wed, 23 Mar 2005 08:23:21 -0600
Subject: Errorfunction
Message-ID: <20050323082321-0600@page.axiom-developer.org>

\begin{axiom}
integrate(exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

From unknown Sat May 21 12:49:39 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:49:39 -0500
Subject: 
Message-ID: <20050521124939-0500@page.axiom-developer.org>

\begin{axiom}
int(x,x)
\end{axiom}

From unknown Sat May 21 12:50:47 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:50:47 -0500
Subject: 
Message-ID: <20050521125047-0500@page.axiom-developer.org>

\begin{axiom}
integrate(x,x)
\end{axiom}

From unknown Sat May 21 12:51:59 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:51:59 -0500
Subject: 
Message-ID: <20050521125159-0500@page.axiom-developer.org>

\begin{axiom}
axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

From unknown Sat May 21 12:52:20 -0500 2005
From: unknown
Date: Sat, 21 May 2005 12:52:20 -0500
Subject: 
Message-ID: <20050521125220-0500@page.axiom-developer.org>

\begin{axiom}
integrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)
\end{axiom}

The answer should be:
\begin{equation}
15\,{\frac {\sqrt {\pi }}{\sqrt {\pi}}}
\end{equation}


From unknown Thu Aug 25 05:57:53 -0500 2005
From: unknown
Date: Thu, 25 Aug 2005 05:57:53 -0500
Subject: integrate(exp(x)/x^2)
Message-ID: <20050825055753-0500@page.axiom-developer.org>

Axiom does not perform the integration (while it perform the integration of exp(x)/x ), but the integration can be given in terms of Ei(x)

integrate(exp(x)/x^2,x)  -->  Ei(x)-exp(x)/x 

From unknown Sat Oct 22 19:04:53 -0500 2005
From: unknown
Date: Sat, 22 Oct 2005 19:04:53 -0500
Subject: 
Message-ID: <20051022190453-0500@page.axiom-developer.org>

int(sqrt(x), x)

---

integrate(exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)

(6)

int(x,x)

   There are no exposed library operations named int but there are 5 
      unexposed operations with that name. Use HyperDoc Browse or issue
                               )display op int
      to learn more about the available operations.

   Cannot find a definition or applicable library operation named int 
      with argument type(s) 
                                 Variable(x)
                                 Variable(x)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

integrate(x,x)

(7)

axiomintegrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)

   There are no library operations named axiomintegrate 
      Use HyperDoc Browse or issue
                           )what op axiomintegrate
      to learn if there is any operation containing " axiomintegrate " 
      in its name.

   Cannot find a definition or applicable library operation named 
      axiomintegrate with argument type(s) 
                             Expression(Integer)
                 SegmentBinding(OrderedCompletion(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

integrate(x^6*exp(-x^2/2)/sqrt(%pi*2),x=%minusInfinity..%plusInfinity)

(8)

The answer should be:

(9)

Axiom does not perform the integration (while it perform the integration of exp(x)/x ), but the integration can be given in terms of Ei(x)

integrate(exp(x)/x^2,x) --> Ei(x)-exp(x)/x

int(sqrt(x), x)