This example was reported to me after a Calculus II student encountered this ridiculous result:

f := (k+5)/sqrt(k^7+k^2):
Int( f, k=1..infinity ):
% = value( % );
             /infinity                              
            |              k + 5                    
            |          -------------- dk = -infinity
            |                   (1/2)               
           /1          / 7    2\                    
                       \k  + k /                    

This result is absurd because the integrand is positive, and decays to zero fast enough for this integral to exist.

Let's try to dig a little deeper to how Maple might be coming to this result:

f2 := simplify( f ) assuming positive;
                             k + 5     
                        ---------------
                                  (1/2)
                          / 5    \     
                        k \k  + 1/     
int( f2, k );
           /[1  1]  [6]    5\
k hypergeom|[-, -], [-], -k |
           \[5  2]  [5]     /

   + 

                                             /              (1/2)\
                         (1/2)       (1/2)   |1   1 / 5    \     |
  (-2 ln(2) + 5 ln(k)) Pi      - 2 Pi      ln|- + - \k  + 1/     |
                                             \2   2              /
  ----------------------------------------------------------------
                                (1/2)                             
                              Pi                                  

qq := int( f2, k=1..N ) assuming positive;
    (3/2)    /1   \    /1   \                                 
  Pi      csc|- Pi| sec|- Pi|                                 
             \5   /    \5   /   2          /[3   1]  [13]    \
- --------------------------- + - hypergeom|[--, -], [--], -1|
            /4\      / 7\       3          \[10  2]  [10]    /
     5 GAMMA|-| GAMMA|--|                                     
            \5/      \10/                                     

         /1   1  (1/2)\            /[1  1]  [6]    5\            
   + 2 ln|- + - 2     | + hypergeom|[-, -], [-], -N | N + 5 ln(N)
         \2   2       /            \[5  2]  [5]     /            

         /              (1/2)\
         |1   1 / 5    \     |
   - 2 ln|- + - \N  + 1/     |
         \2   2              /
limit( qq, N=infinity );
         /    (3/2)    /1   \    /1   \
         |  Pi      csc|- Pi| sec|- Pi|
         |             \5   /    \5   /
    limit|- ---------------------------
         |            /4\      / 7\    
         |     5 GAMMA|-| GAMMA|--|    
         \            \5/      \10/    

         2          /[3   1]  [13]    \       /1   1  (1/2)\
       + - hypergeom|[--, -], [--], -1| + 2 ln|- + - 2     |
         3          \[10  2]  [10]    /       \2   2       /

                  /[1  1]  [6]    5\            
       + hypergeom|[-, -], [-], -N | N + 5 ln(N)
                  \[5  2]  [5]     /            

                                                \
             /              (1/2)\              |
             |1   1 / 5    \     |              |
       - 2 ln|- + - \N  + 1/     |, N = infinity|
             \2   2              /              |
                                                |
                                                /
value( % );
         /    (3/2)    /1   \    /1   \
         |  Pi      csc|- Pi| sec|- Pi|
         |             \5   /    \5   /
    limit|- ---------------------------
         |            /4\      / 7\    
         |     5 GAMMA|-| GAMMA|--|    
         \            \5/      \10/    

         2          /[3   1]  [13]    \       /1   1  (1/2)\
       + - hypergeom|[--, -], [--], -1| + 2 ln|- + - 2     |
         3          \[10  2]  [10]    /       \2   2       /

                  /[1  1]  [6]    5\            
       + hypergeom|[-, -], [-], -N | N + 5 ln(N)
                  \[5  2]  [5]     /            

                                                \
             /              (1/2)\              |
             |1   1 / 5    \     |              |
       - 2 ln|- + - \N  + 1/     |, N = infinity|
             \2   2              /              |
                                                |
                                                /
evalf( %% );
                               0.

This result is equally suspect, for the same reason. There is definitely a positive area under this curve, so the integral can't be zero - it should be either real and positive or infinite.

From a plot of q2 it is pretty clear that the value of this improper integral is about 2.366

This appears to be a bug. Can anyone give an explanation that disputes my conjecture that this is an error?

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu