Improper Integrals

     ¥                             b
    ò  f(x) dx  =  lim       ò  f(x) dx
   a                 b®¥   a

      b                                    b
    ò  f(x) dx  =  lim            ò  f(x) dx
 -¥                   a® -¥   a

    ¥                                    c                               b
    ò  f(x) dx  =  lim            ò  f(x) dx  +  lim       ò  f(x) dx
 -¥                   a® -¥   a                   b®¥  c
 

II.  Integrals of functions that become infinite at a point within the interval of integration are
      improper integrals.

      If f(x) is continuous on (a,b], then
            b                                    b
          ò  f(x) dx  =  lim            ò  f(x) dx
       a                   c® a⁺       c

      If f(x) is continuous on [a,b), then
            b                                    c
          ò  f(x) dx  =  lim            ò  f(x) dx
       a                   c® b^-        a

      If f(x) is continuous on [a,c) and (c,b], then
            b                              c                                     b
          ò  f(x) dx  =  lim      ò  f(x) dx  +   lim            ò  f(x) dx
       a                   c® b^-  a                     c® a⁺    c

III.  Direct Comparison Test

        Let f(x) < g(x) for all x in the interval.  Then

          ¥                                 ¥
        ò  f(x) dx  converges if  ò g(x) dx  converges.
       a                                   a
 

          ¥                                ¥
        ò  g(x) dx  diverges  if  ò  f(x) dx   diverges.
      a                                  a
 

IV.  Limit Comparison Test

       If f and g are positive functions on [a, ¥)  and  if

        lim       f(x)   =  L ,     L > 0
        x®¥  g(x)

        then

          ¥                     ¥
        ò  f(x) dx  and  ò g(x) dx  either both converge or both diverge.
       a                     a

       In other words, if both functions grow at the same rate, they act alike.