Recently I stumbled upon someone who said he wanted to understand why
$\arctan x = \int\dfrac{dx}{1+x^2}$
At first I was confused. This is an easy result in any integral calculus course. But then he explained that although he understood the proof, he wanted to understand it "intuitively". He wanted to see why it was in terms of arclength and addition and subtraction.
My question is: Is there an "intuitive" way to explain this identity?