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.
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?
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