sinc函数的积分

• 求sinc函数的积分：
  $$I={\int }_{-\infty }^{+\infty }\frac{sin(x)}{x}dx$$
• 解法如下：
  $$易知有{\int }_0^{+\infty }{e}^{-xt}dt=\frac{1}{x}$$
  $$\begin{array}{rl}I& ={\int }_{-\infty }^{+\infty }\frac{sin(x)}{x}dx\\ & =2{\int }_0^{+\infty }\frac{sin(x)}{x}dx\\ & =2{\int }_0^{+\infty }sin(x)({\int }_0^{+\infty }{e}^{-xt}dt)dx\\ & =2{\int }_0^{+\infty }[{\int }_0^{+\infty }sin(x)e^{-xt}dx]dt\\ & =2{\int }_0^{+\infty }\frac{1}{1+t^2}dt(此步可由分部积分得到)\\ & =\pi \end{array}$$