How do I find the integral of
F(x) = the integral of 1/1+t^2 dt from 0 to x + the integral of 1/1+t^2 dt from 0 to 1/x?
and how do I determine that F(x) is constant on (-infinity,0) and constant on (0,infinity)?
Any help appreciated!
Since (F+ G)'= F+ G', you could also show this function is a constant by differentiating each of the integrals separately and adding. By the "Fundamental Theorem of Calculus" the derivative of the first integral is just . By the "Fundamental Theorem of Calculus", together with the chain rule, the derivative of the second integral is . Those clearly add to 0, for all x (except x= 0 where is not defined), so the function is a constant.