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2019/01/29

a formula for π 3

There are many equations of \pi .

 In first post of \pi , we see
\pi=\int_{-\infty}^{\infty}\frac{1}{1+x^2}dx   

If we shorten the interval of integration from (-\infty,\infty) to [0,1] ,then
  \int_0^1\frac{1}{1+x^2}dx=\frac{\pi}{4} . 

Because, put x=\tan \theta . We know
\frac{dx}{d\theta}=\frac{1}{\cos^2\theta} 
and if x=0 , then \theta=0, and if x\rightarrow 1 , then \theta\rightarrow \frac{\pi}{4} .
Therefore, 
\int_0^1\frac{1}{1+x^2}dx=\int_0^{\pi/4}\frac{1}{1+\tan^2\theta}\frac{1}{\cos^2\theta}d\theta=\frac{\pi}{4} . 

We will get easily
  \int_{-1}^1\frac{1}{1+x^2}dx=\frac{\pi}{2} . 
Thus,
  \int_1^{\infty}\frac{1}{1+x^2}dx=\int_{-\infty}^{-1}\frac{1}{1+x^2}dx=\frac{\pi}{4} .













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