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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?

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I believe you mean $1+x^2$ in the denominator. – Akhil Mathew Aug 20 '10 at 23:45
Just a thought: it's probably easier to find an intuitive explanation of the equivalent d/dx arctan x = 1/(1+x^2). – Michael Lugo Aug 21 '10 at 0:32
@Akjil Thank you. I knew I would make a typo somewhere on my first post. – AnonymousCoward Aug 22 '10 at 4:48
As the waitress said, "plus a constant!".… – Arturo Magidin Jan 5 '11 at 15:36

7 Answers 7

up vote 18 down vote accepted

The geometric picture is as follows. Let $O = (0, 0), A = (1, 0), X = (1, x)$. Then $\arctan x = \angle AOX$. We want to understand why $\angle AOY \approx \arctan x + \frac{h}{1 + x^2}$ where $Y = (1, x + h)$ and $h$ is small; equivalently, we want to understand why $\angle XOY \approx \frac{h}{1 + x^2}$. Since this angle is small, we equivalently want to understand why $\sin \angle XOY \approx \frac{h}{1 + x^2}$.

Now $\triangle XOY$ evidently has area $\frac{h}{2}$. On the other hand, it has area $\frac{1}{2} |OX| |OY| \sin \angle XOY$ where $|OX| = \sqrt{1 + x^2}$ and $|OY| \approx |OX|$. The result follows.

(The derivative follows, anyway. The integral follows by dividing up $AX$ into little pieces and drawing a bunch of lines to $O$, then summing up all of the contributions.)

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+1 for the geometric explanation. Would you mind inserting the geometric figure? – Américo Tavares Aug 25 '10 at 11:17

Stereographic projection offers a geometric interpretation of this formula. In fact the factor $(1+x^2)^{-1}$ is the metric of the circle in the stereographic projection coordinate. The construction is as follows:

Take a circle of radius 1/2, call the angle of (the radius vector of) a general point on the circle with respect to the radius to the south pole $2\theta$. Connect the point with the north pole and call the distance of the point of intersection of the extension of this chord with the tangent at the south pole $x$. One can parameterize the points on the circle by the coordinate $x$ which is called the stereographic projection coordinate. It is easy to see that $ x = tan(\theta)$. The length of an arc of the circle between $\theta_1$ and$\theta_2$ (having the stereographic projection coordinates $x_1$ and $x_2$) is given by:

$s = 2 \int_{\theta_1}^{\theta_2} d\theta = 2\int_{x_1}^{x_2} \frac{dx}{1+x^2}$

This construction has further nice properties

  1. The group of plane rotations $SO(2)$ acts on $x$ by a Möbius transformation.

  2. This construction generalizes to spheres in higher dimensions.

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alt text

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This was actually my question originally. To put the demand for intuitiveness in starker terms: what does the arc of a circle have to do with the area of a square and the operation of reciprocation?

If someone can show me a geometric figure with area $1 \over 1 + x^2$ that might be a good start.

Or, they can explain how this sequence

  1. Times a number by itself;
  2. Add one to it;
  3. Flip that result

leads to "a little bit of" circular arc.

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@Lao Tzu: have you read my answer? The 1/(1+x^2) term basically comes from the Pythagorean theorem, and it doesn't measure an area but an angle. – Qiaochu Yuan Aug 25 '10 at 14:34
I've read it, and it's a huge improvement but not all the way to where I want to go. Thanks for going into the geometric detail. – isomorphismes Aug 26 '10 at 6:42
One problem for me is that $\sin x \approx x$ is still too Taylor-y for me to explain to a fifth-grader. – isomorphismes Aug 26 '10 at 6:43
Also, why is it evident that $\Delta XOY$ has area $h \over 2$? Let $x=.0001$ for example. – isomorphismes Aug 26 '10 at 6:48
@Lao Tzu: sin x being approximately x is just the statement that a circle is approximately linear. More precisely, compare the length of a very small arc from (cos x, sin x) to (1, 0) where x is small with the length of the perpendicular from (cos x, sin x) to (cos x, 0). As for the triangle, take XY as the base of the triangle. (PS: in the future to ensure that I am notified of these comments you should begin them with @Qiaochu Yuan.) – Qiaochu Yuan Aug 29 '10 at 7:03

Put $\arctan x=:\alpha$, let $n$ be large, choose points $t_k:=\tan(k\alpha/n)$ $\ (0\leq k\leq n)$ and put $\tau_k:=\sqrt{t_k t_{k-1}}$. Then $$\sum_{k=1}^n {t_k - t_{k-1} \over 1 + \tau_k^2}=\sum_{k=1}^n {t_k - t_{k-1} \over 1 + t_k t_{k-1}}= n \tan(\alpha/n).$$ Here the left side is a Riemann sum for the integral $$\int_0^{\tan\alpha}{dt \over 1+t^2}=\int_0^x{dt \over 1+t^2},$$ and the right side has limit $\alpha=\arctan x$ when $n\to\infty$.

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how is the first equation found? – anon Aug 25 '10 at 6:33
First equality because $\tau_k^2=\tau_k \tau_{k+1}$. Second by the angle addition formula for tangent… – David Speyer Aug 25 '10 at 11:16
PS Nice answer! – David Speyer Aug 25 '10 at 11:17

Since $D \tan(x) = 1 + \tan(x)^2$ and $\tan(\arctan(x)) = x$ we see that:

$$\begin{align} \tan(\arctan(x)) &= x \\ (1 + \tan(\arctan(x))^2 \cdot D \arctan(x) &= 1 \\ D \arctan(x) &= \frac{1}{(1 + x)^2} \\ \end{align}$$

So by the fundamental theorem of calculus we recover the integral.

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Again, this is a proof, but I do not see how it directly provides any intuition. – Qiaochu Yuan Aug 21 '10 at 8:20

The real answer I think involves complex numbers. Factor $1+x^2 = (i-x)(-i-x)$, decompose in partial fractions, get an expression involving logarithms that ends up being $\arctan x$ as described in

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This is a proof (one of my favorites), but I do not immediately see how it relates to the geometric definition of arctan, which I assume is what the OP is asking about. – Qiaochu Yuan Aug 21 '10 at 2:58
"Intuitively" does not mean "geometrically", does it? For me, explaining integrals in terms in logarithms is intuitive, but I'm not a first-year calculus student. :-) – lhf Aug 21 '10 at 11:45
What I mean is, this is a general-purpose algorithm for obtaining the answer. But why would one expect the answer in this case to be arctan and not arctanh or some other expression involving logarithms? – Qiaochu Yuan Aug 21 '10 at 18:30

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