Integral of 1/(1+X^2) Using Geometric Approach – ATan ArcTan

Published by Fudgy McFarlen on June 26, 2014June 26, 2014

I had to do a trigonometric integral the other day and realized I should be able to see the integral at a glance instead of the mechanistic method I used by substituting $x=tan(theta)$

$W={Delta}{theta}*z$

Due to similarity of triangles

${z/1}={{Delta}x}/w$

Substituting

${z/1}={{Delta}x}/{{Delta}{theta}*z}$

${{Delta}x}/z^2={Delta}{theta}$

${{Delta}x}/{1+x^2}={Delta}{theta}$

$int{.}{.}{.}{{Delta}x}/{1+x^2}=int{.}{.}{.}{Delta}{theta}$

$int{.}{.}{.}{{Delta}x}/{1+x^2}={theta}=atan(x)$

Feynman uses this principle of areas of triangles being the same as you travel at a constant velocity as a jumping off point for Kepler's equal area sweep law here in this video:

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Area of the Rectangle Puzzle

Calculus Proof of the Pythagorean Theorem