Evaluate this integral in terms of a and b, where each of a and b is a positive real number.

∞
∫ Arctan(ax)−Arctan(bx) dx
0

I'm thinking this is undefined (except where a=b).  Playing with integrating a=2, b=1 from 0 to 10, 100, 1000, 10000 etc.  Adding a zero increases the integral by more than 1.

The integral formula for arctan has a ln term that seems to grow as well.

The real evidence was to change to the inverse of the function.
∞
∫ Arctan(ax)−Arctan(bx) dx =
0

π/2
∫ (1/a)tan(x)−(1/b)tan(x) dx =
0

              π/2
(1/a-1/b)∫ tan(x) dx
              0

The integral formula for tan(x) is
-ln|cos(x)| which is undefined at π/2

Posted by Jer on 2014-04-04 22:02:57