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例題 次の広義積分を求めよ．

$\displaystyle I = \int_{-\infty}^{\infty}\frac{dx}{1+x^{2}} $

解答

$\displaystyle I$ $\displaystyle =$ $\displaystyle \lim_{{a \rightarrow -\infty}, {b \rightarrow \infty}}\int_{a}^{b}\frac{dx}{1+x^{2}} = \left[\tan^{-1}{x}\right]_{-\infty + }^{\infty - }$  
  $\displaystyle =$ $\displaystyle \lim_{x \rightarrow \infty} \tan^{-1}{x} - \lim_{x \rightarrow - \infty} \tan^{-1}{x} = \frac{\pi}{2} + \frac{\pi}{2} = \pi$