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

  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 [pmath]x=tan(theta)[/pmath]

[pmath] W={Delta}{theta}*z [/pmath]

Due to similarity of triangles

[pmath] {z/1}={{Delta}x}/w [/pmath]

Substituting

[pmath] {z/1}={{Delta}x}/{{Delta}{theta}*z} [/pmath]

[pmath] {{Delta}x}/z^2={Delta}{theta} [/pmath]

[pmath] {{Delta}x}/{1+x^2}={Delta}{theta} [/pmath]

[pmath] int{.}{.}{.}{{Delta}x}/{1+x^2}=int{.}{.}{.}{Delta}{theta} [/pmath]

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

Related Topic

• Area of the Rectangle Puzzle

Calculus Proof of the Pythagorean Theorem