reduce-historical/r36/xlog/DEFINT.LOG

REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ...



*** ci already defined as operator 

*** si already defined as operator 
% Test cases for definite integration.

int(x/(x+2),x,2,6);


2*( - log(2) + 2)


int(sin x,x,0,pi/2);


1


int(log(x),x,1,5);


5*log(5) - 4


int((1+x**2/p**2)**(1/2),x,0,p);


 p*(sqrt(2) + log(sqrt(2) + 1))
--------------------------------
               2


int(x**9+y+y**x+x,x,0,2);


                               2
 10*log(y)*y + 522*log(y) + 5*y  - 5
-------------------------------------
              5*log(y)



% Collected by Kerry Gaskell, ZIB, 1993/94.

int(x^2*log(1+x),x,0,infinity);


     2
int(x *log(1 + x),x,0,infinity)


int(x*e^(-1/2x),x,0,infinity);


4


int(x/4*e^(-1/2x),x,0,infinity);


1


int(sqrt(2)*x^(1/2)*e^(-1/2x),x,0,infinity);


2*sqrt(pi)


int(x^(3/2)*e^(-x),x,0,infinity);


 3*sqrt(pi)
------------
     4


int(sqrt(pi)*x^(3/2)*e^(-x),x,0,infinity);


 3*pi
------
  4


int(x*log(1+1/x),x,0,infinity);


               1
int(x*log(1 + ---),x,0,infinity)
               x


int(si(1/x),x,0,infinity);


        1
int(si(---),x,0,infinity)
        x


int(cos(1/x),x,0,infinity);


         1
int(cos(---),x,0,infinity)
         x


int(sin(x^2),x,0,infinity);


 sqrt(pi)*sqrt(2)
------------------
        4


int(sin(x^(3/2)),x,0,infinity);


           2/3        5
 sqrt(pi)*2   *gamma(---)
                      6
--------------------------
                2
       3*gamma(---)
                3


int(besselj(2,x),x,0,infinity);


1


int(besselj(2,y^(5/4)),y,0,infinity);


    4/5        7
 2*2   *gamma(---)
               5
-------------------
            8
   5*gamma(---)
            5


int(x^(-1)*besselj(2,sqrt(x)),x,0,infinity);


1


int(bessely(2,x),x,0,infinity);


int(bessely(2,x),x,0,infinity)


int(x*besseli(2,x),x,0,infinity);


int(x*besseli(2,x),x,0,infinity)


int(besselk(0,x),x,0,infinity);


 pi
----
 2


int(x^2*besselk(2,x),x,0,infinity);


 3*pi
------
  2


int(sinh(x),x,0,infinity);


int(sinh(x),x,0,infinity)


int(cosh(2*x),x,0,infinity);


int(cosh(2*x),x,0,infinity)


int(-3*ei(-x),x,0,infinity);


3


int(x*shi(x),x,0,infinity);


int(x*shi(x),x,0,infinity)


int(x*fresnel_c(x),x,0,infinity);


int(x*fresnel_c(x),x,0,infinity)


int(x^3*e^(-2*x),x,0,infinity);


 3
---
 8


int(x^(-1)*sin(x/3),x,0,infinity);


 pi
----
 2


int(x^(-1/2)*sin(x),x,0,infinity);


 sqrt(pi)*sqrt(2)
------------------
        2


int(2*x^(-1/2)*cos(x),x,0,infinity);


sqrt(pi)*sqrt(2)


int(sin x + cos x,x,0,infinity);


int(sin(x) + cos(x),x,0,infinity)


int(ei(-x) + sin(x^2),x,0,infinity);


 sqrt(pi)*sqrt(2) - 4
----------------------
          4


int(x^(-1)*(sin (-2*x) + sin(x^2)),x,0,infinity);


  - pi
-------
   4


int(x^(-1)*(cos(x/2) - cos(x/3)),x,0,infinity);


        3
 - log(---)
        2


int(x^(-1)*(cos x - cos(2*x)),x,0,infinity);


log(2)


int(x^(-1)*(cos(x) - cos(x)),x,0,infinity);


0


int(2,x,0,infinity);


int(2,x,0,infinity)


int(cos(x)*si(x),x,0,infinity);


int(cos(x)*si(x),x,0,infinity)


int(2*cos(x)*e^(-x),x,0,infinity);


1


int(x/2*cos(x)*e^(-x),x,0,infinity);


0


int(x^2/4*cos(x)*e^(-2*x),x,0,infinity);


  1
-----
 125


int(1/(2*x)*sin(x)*e^(-3*x),x,0,infinity);


       1
 atan(---)
       3
-----------
     2


int(3/x^2*sin(x)*e^(-x),x,0,infinity);


     3            - x
int(----*sin(x)*e    ,x,0,infinity)
      2
     x


int(cos(sqrt(x))*e^(-x),x,0,infinity);


                  i        1/4
 sqrt( - pi)*erf(---) + 2*e
                  2
-------------------------------
               1/4
            2*e


int(e^(-x)*besselj(2,x),x,0,infinity);


  - 2*sqrt(2) + 3
------------------
     sqrt(2)


int(cos(x^2)*e^(-x),x,0,infinity);


               1              1          1                  1        1
(pi*( - 2*cos(---)*fresnel_s(---) + cos(---) + 2*fresnel_c(---)*sin(---)
               4              4          4                  4        4

             1
      - sin(---)))/(2*sqrt(pi)*sqrt(2))
             4


int(erf(x)*e^(-x),x,0,infinity);


 1/4          1
e   *( - erf(---) + 1)
              2


int(besseli(2,x)*e^(-x),x,0,infinity);


                    - 1                            1
2*hypergeometric({------},{},1) + hypergeometric({---},{},1) - 2
                    2                              2


int(e^(-x^2)*cos(x),x,0,infinity);


 sqrt(pi)
----------
     1/4
  2*e


int(x^(-1)*sin(x)*cos(x),x,0,infinity);


 pi
----
 4


int(x^(-1)*sin(x)*cos(2*x),x,0,infinity);


0


int(x^(-1)*sin(x)*cos(x/2),x,0,infinity);


 pi
----
 2


int(e^x,x,0,infinity);


     x
int(e ,x,0,infinity)


int(e^(-x^2 - x),x,0,infinity);


  1/4             1
 e   *pi*( - erf(---) + 1)
                  2
---------------------------
        2*sqrt(pi)


int(e^(-(x+x^2+1)),x,0,infinity);


  1/4             1
 e   *pi*( - erf(---) + 1)
                  2
---------------------------
       2*sqrt(pi)*e


int(e^(-(x+1/x)^2),x,0,infinity);


 sqrt(pi)
----------
      4
   2*e


int(e^(-(x+2))*sin(x),x,0,infinity);


  1
------
    2
 2*e


int(-3*x*e^(-1/2x),x,0,infinity);


-12


int(x*e^(-1/2*x^2),x,0,infinity);


1


int(x^2*besselj(2,x),x,0,infinity);


     2
int(x *besselj(2,x),x,0,infinity)


int(x*besselk(1,x),x,0,infinity);


 pi
----
 2


int(-3*ei(-x),x,0,infinity);


3


int(x^3*e^(-2*x^2),x,0,infinity);


 1
---
 8


int(sqrt(2)/2*x^(-3/2)*sin x,x,0,infinity);


sqrt(pi)


int(x^(-1)*(sin(-2*x) + sin(x^2)),x,0,infinity);


  - pi
-------
   4


int(x^(-1)*(cos(3*x) - cos(x/2)),x,0,infinity);


 - log(6)


int(x^(-1)*(sin x - sin(2*x)),x,0,infinity);


0


int(1/x*sin(x)*e^(-3*x),x,0,infinity);


      1
atan(---)
      3


int(sin(x)*e^(-x),x,0,infinity);


 1
---
 2


int(x^(-1)*sin(x)*cos(x),x,0,infinity);


 pi
----
 4


int(e^(1-x)*e^(2-x^2),x,0,infinity);


  1/4  3             1
 e   *e *pi*( - erf(---) + 1)
                     2
------------------------------
          2*sqrt(pi)


int(e^(-1/2x),x,0,y);


     y/2
 2*(e    - 1)
--------------
      y/2
     e


int(si(x),x,0,y);


cos(y) + si(y)*y - 1


int(besselj(2,x^(1/4)),x,0,y);


              1/4
 4*besselj(3,y   )*y
---------------------
         1/4
        y


int(x*besseli(2,x),x,0,y);


besseli(1,y)*y - 2*besseli(0,y) + 2


int(x^(3/2)*e^(-x),x,0,y);


             y
 3*sqrt(pi)*e *erf(sqrt(y)) - 4*sqrt(y)*y - 6*sqrt(y)
------------------------------------------------------
                            y
                         4*e


int(sinh(x),x,0,y);


  2*y      y
 e    - 2*e  + 1
-----------------
         y
      2*e


int(cosh(2*x),x,0,y);


  4*y
 e    - 1
----------
     2*y
  4*e


int(x*shi(x),x,0,y);


     2*y      2*y      y         2
  - e   *y + e    + 2*e *shi(y)*y  - y - 1
-------------------------------------------
                      y
                   4*e


int(x^2*e^(-x^2),x,0,y);


            2
           y
 sqrt(pi)*e  *erf(y) - 2*y
---------------------------
               2
              y
           4*e


int(x^(-1)/2*sin(x),x,0,y);


 si(y)
-------
   2


int(sin x + cos x,x,0,y);


 - cos(y) + sin(y) + 1


int(sin x + sin(-2*x),x,0,y);


 cos(2*y) - 2*cos(y) + 1
-------------------------
            2


int(sin(n*x),x,0,y);


  - cos(n*y) + 1
-----------------
        n


int(heaviside(x-1),x,0,y);


heaviside(y - 1)*(y - 1)



% Tests of transformations defined in defint package.

laplace_transform(1,x);


 1
---
 s


laplace_transform(x,x);


 1
----
  2
 s
 

laplace_transform(x^a/factorial(a),x);


  1
------
  a
 s *s


laplace_transform(x,e^(-a*x),x);


        1
-----------------
  2            2
 a  + 2*a*s + s


laplace_transform(x^k,e^(-a*x),x);


      gamma(k + 1)
-------------------------
        k            k
 (a + s) *a + (a + s) *s
      

laplace_transform(cosh(a*x),x);


   - s
---------
  2    2
 a  - s
 

laplace_transform(1/(2*a^3),sinh(a*x)-sin(a*x),x);


   - 1
---------
  4    4
 a  - s


laplace_transform(1/(a^2),1-cos(a*x),x);


     1
-----------
  2      3
 a *s + s
        

laplace_transform(1/(b^2-a^2),cos(a*x)-cos(b*x),x);


             s
----------------------------
  2  2    2  2    2  2    4
 a *b  + a *s  + b *s  + s


laplace_transform(besselj(0,2*sqrt(k*x)),x);


   1
--------
  k/s
 e   *s
    

laplace_transform(Heaviside(x-1),x);


  1
------
  s
 e *s


laplace_transform(1/x,sin(k*x),x);


      k
atan(---)
      s


laplace_transform(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x);


     2  2              2  2
    k *s              k *s
 - e     *erf(k*s) + e
   

laplace_transform(1/k,e^(-k^2/(4*x)),x);


 besselk(1,sqrt(s)*k)
----------------------
       sqrt(s)


laplace_transform(2/(sqrt(pi*x)),besselk(0,2*sqrt(2*k*x)),x);


  k/s            k
 e   *besselk(0,---)
                 s
---------------------
       sqrt(s)
   

hankel_transform(x,x);


         n + 4
  gamma(-------)
           2
-------------------
        n - 2    2
 gamma(-------)*s
          2
  

Y_transform(x,x);


          - n + 4          n + 4
  gamma(----------)*gamma(-------)
            2                2
-------------------------------------
         - n + 3          n - 1    2
 gamma(----------)*gamma(-------)*s
           2                2
       

K_transform(x,x);


         - n + 4          n + 4
 gamma(----------)*gamma(-------)
           2                2
----------------------------------
                  2
               2*s
       

struveh_transform(x,x);


          - n - 3          n + 5
  gamma(----------)*gamma(-------)
            2                2
-------------------------------------
         - n - 2          n - 2    2
 gamma(----------)*gamma(-------)*s
           2                2


fourier_sin(e^(-x),x);


   s
--------
  2
 s  + 1
  

fourier_sin(sqrt(x),e^(-1/2*x),x);


           3*atan(2*s)
    2*sin(-------------)*pi
                2
--------------------------------
              2     3/4
 sqrt(pi)*(4*s  + 1)   *sqrt(2)
      

fourier_sin(1/x,e^(-a*x),x);


      s
atan(---)
      a
    

fourier_sin(x^k,x);


  k/2         - k          k
 4   *gamma(------)*gamma(---)*k
              2            2
---------------------------------
         k  k
      4*s *2 *gamma( - k)*s
     

fourier_sin(1/(b-a),(e^(-a*x)-e^(-b*x)),x);


         a*s + b*s
----------------------------
  2  2    2  2    2  2    4
 a *b  + a *s  + b *s  + s


fourier_sin(besselj(0,a*x),x);


                2    2
             - a  + s
 heaviside(------------)
                 2
                a
-------------------------
             2    2
    sqrt( - a  + s )


fourier_sin(1/sqrt(pi*x),cos(2*sqrt(k*x)),x);


                      k                          k
 sqrt(s)*sqrt(2)*cos(---) - sqrt(s)*sqrt(2)*sin(---)
                      s                          s
-----------------------------------------------------
                         2*s


fourier_sin(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x);


 sqrt( - pi)*erf(i*k*s)
------------------------
               2  2
              k *s
    sqrt(pi)*e


fourier_cos(e^(-1/2x),x);


    2
----------
    2
 4*s  + 1
       

fourier_cos(x,e^(-x),x);


       2
    - s  + 1
---------------
  4      2
 s  + 2*s  + 1
        

fourier_cos(x,e^(-1/2*x^2),x);


                                  2
                    i*s          s /2
 sqrt( - pi)*erf(---------)*s + e    *sqrt(2)
                  sqrt(2)
----------------------------------------------
                  2
                 s /2
                e    *sqrt(2)
  

fourier_cos(2*x^2,e^(-1/2x),x);


              2
       - 384*s  + 32
---------------------------
     6       4       2
 64*s  + 48*s  + 12*s  + 1
 

fourier_cos(x,e^(-a*x),x);


       2    2
      a  - s
-------------------
  4      2  2    4
 a  + 2*a *s  + s
      

fourier_cos(x^n,e^(-a*x),x);


           s             s
 cos(atan(---)*n + atan(---))*gamma(n + 1)
           a             a
-------------------------------------------
              2    2 (n + 1)/2
            (a  + s )
    

fourier_cos(1/x,sin(k*x),x);


       2    2
 sign(k  - s )*pi + pi
-----------------------
           4


fourier_cos(1/sqrt(pi*x),cos(2*sqrt(k*x)),x);


                      k                          k
 sqrt(s)*sqrt(2)*cos(---) + sqrt(s)*sqrt(2)*sin(---)
                      s                          s
-----------------------------------------------------
                         2*s
   

fourier_cos(1/(k*sqrt(pi)),e^(-x^2/(4*k^2)),x);


   1
--------
   2  2
  k *s
 e


fourier_cos(1/(pi*x),sin(2*k*sqrt(x)),x);


        2             2
       k             k
intfc(----) + intfs(----)
       s             s
       

fourier_cos(1/(sqrt(pi*x)),e^(-2*k*sqrt(x)),x);


                    2               2                   2
                   k               k                   k
( - 2*sqrt(s)*cos(----)*fresnel_s(----) + sqrt(s)*cos(----)
                   s               s                   s

                          2         2                   2
                         k         k                   k
  + 2*sqrt(s)*fresnel_c(----)*sin(----) - sqrt(s)*sin(----))/(sqrt(2)*s)
                         s         s                   s


laplace_transform(x^n/factorial(n)*e^(-a*x),x);


            1
-------------------------
        n            n
 (a + s) *a + (a + s) *s


laplace_transform(1/(2*a^2)*(cosh(a*x)-cos(a*x)),x);


   - s
---------
  4    4
 a  - s


laplace_transform(k*a^k/x*besselj(k,a*x),x);


          2*k
         a
----------------------
        2    2      k
 (sqrt(a  + s ) + s)
    

fourier_sin(1/x*e^(-3*x),x);


      s
atan(---)
      3
    

fourier_sin(1/(pi*x)*sin(2*k*sqrt(x)),x);


        2             2
       k             k
intfc(----) - intfs(----)
       s             s


fourier_cos(x^n*e^(-a*x),x);


           s             s
 cos(atan(---)*n + atan(---))*gamma(n + 1)
           a             a
-------------------------------------------
              2    2 (n + 1)/2
            (a  + s )
    

fourier_cos(1/(k*sqrt(pi))*e^(-x^2/(4*k^2)),x);


   1
--------
   2  2
  k *s
 e


end;
(TIME:  defint 163620 182910)