1. The problem statement, all variables and given/known data Consider the integral INT -2x/(1+x^2) dx FROM -INF TO INF (The attached TheIntegral.jpg file shows this in a more aesthetically-pleasing manner.) If the integral is divergent, state that it is so. Otherwise, evaluate the integral. 2. Relevant equations Integration. U substitution. 3. The attempt at a solution I turned the integral the problem gives into: lim t-> inf INT -2x/(1+x^2) dx FROM -t TO t Letting u = 1+x^2, du/dx = 2x, dx = du/2x Lower limit: u = 1 + (-t)^2 = 1 + t^2, Upper limit: u = 1 + (t)^2 = 1 + t^2 lim t-> inf INT -1/u du FROM (1 + t^2) TO (1 + t^2) lim t-> inf [-ln|u|] FROM (1 + t^2) TO (1 + t^2) = 0 Using Wolfram Alpha, I know the integral is divergent but, when I use Maxima, I get Principal Value: 0 (and, I don't know what means, but it's what I get). In addition to me agreeing with the value that Maxima gives, I also used another piece of software called Kmplot which is how I got the image I attached in this thread, TheGraph.jpg, which shows a symmetric graph with areas that, to my intuition at least, seem to cancel out. Could someone please explain to me why this integral is a divergent integral rather than one that converges to a value of 0?