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} .
0 件のコメント:
コメントを投稿