Question

how do you solve arctan(x) + x/(x^2+1)?

Answer 1

Given f(x)=x arctan(x) then f ' (x)= x/1+x^2 +arctan(x) so how can I mathematically solve f '(x)=0?

Answer 2

Not sure there is a pure mathematical way besides logic.
\[\frac{ (x^{2}+1)arctanx + x }{ x^{2}+1 }\]And thatsif I make it into one fraction. So in order for the top to be 0, (x^2+1)arctan has to be the same value, but a different sign of x. Well, if you start throwing in common values into arctan, its going to spit out radian measures. Without some sort of approximation method, thered be no way to see if some random radian measure beyond the common ones would work, but certain no common one would suffice. As soon as you plug in something like sqrt(3), arctan spits out pi/3 and its obvious quickly that wouldnt work. The idea is that the answer is 0, because without some sort of other method, you can only try and logically figure it out. The answer truly is 0 by the way, I made sure there were no other weird answers.