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Text File  |  2002-05-15  |  1.7 KB  |  62 lines
  1. LOAD("int_misc.mth")
  2.  
  3. ;Removing discontinuities at x=-1 and x=1
  4. MAKECONT(x^2*SIGN(x^2-1),x,[-1,1])
  5.  
  6. ;Fail to remove the discontinuity, because of the lack of side limits
  7. MAKECONT(SIN(1/x),x,[0])
  8.  
  9.  
  10. ;DERIVE finds a discontinuous antiderivative
  11. INT(ASIN(2*x/(x*x+1)),x)
  12.  
  13. ;The correct (continuous) antiderivative
  14. CONT_INT(ASIN(2*x/(x*x+1)),x,[-1,1])
  15.  
  16. ;The result is wrong!
  17. INT(ASIN(2*x/(x*x+1)),x,0,SQRT(3))
  18.  
  19. ;Now it is OK
  20. CONT_INT(ASIN(2*x/(x*x+1)),x,[-1,1],0,SQRT(3))
  21.  
  22. ;Fail to compute the antiderivative
  23. INT(EXP(ASIN(x)),x)
  24.  
  25. ;Trying with INTSUBST
  26. INTSUBST(EXP(ASIN(x)),x,ASIN(x))
  27.  
  28. ;Fail to compute the antiderivative
  29. INT((x+1)/(SQRT(x*x+x+1)+2*x+3),x)
  30.  
  31. ;Using the first Euler substitution
  32. INTSUBST((x+1)/(SQRT(x*x+x+1)+2*x+3),x,x*x+x+1="euler1")
  33.  
  34. ;Giving both the direct and inverse substitutions
  35. INTSUBST((x+1)/(SQRT(x*x+x+1)+2*x+3),x,SQRT(x*x+x+1)=x+t,t,x*x+x+1=(x+t)^2)
  36.  
  37. ;Using the second Euler substitution
  38. INTSUBST((x+1)/(SQRT(x*x+x+1)+2*x+3),x,x*x+x+1="euler2")
  39.  
  40. ;Using the third Euler substitution
  41. INTSUBST(SQRT(x*x-5*x+4),x,x*x-5*x+4="euler3")
  42.  
  43. ;A definite integral
  44. INTSUBST((x+1)/(SQRT(x*x+x-6)+2*x+3),[x,2,3],x*x+x-6="euler1")
  45.  
  46. ;An improper integral
  47. INTSUBST(1/(x*SQRT(x^2+x+1) + 2*x + 1), [x, 0, inf], x^2 + x + 1 = "euler1")
  48.  
  49. ;INTSUBST cannot automatically compute the inverse substituion 
  50. INTSUBST((x+1)/(SQRT(x*x+x+1)+2*x+3),x,SQRT(x*x+x+1)=x+t,t)
  51.  
  52. ;Thank you!
  53. "":='~
  54. ["                Ok, end of the demo!               ";~
  55. " ";~
  56. "Author:  Valeriu Anisiu,    anisiu@math.ubbcluj.ro  ";~
  57. "address: Faculty of Mathematics and Computer Science";~
  58. "         Babes-Bolyai University                    ";~
  59. "         Kogalniceanu 1 Street                      ";~
  60. "         3400 Cluj-Napoca, Romania                  "]
  61.  
  62.