计算反常积分∫上面是正无穷,下面是负无穷,dx/1+x^2

问题描述:

计算反常积分∫上面是正无穷,下面是负无穷,dx/1+x^2

原函数为arctanx,原式=arctan(+无穷大)-arctan(-无穷大)=pi(即圆周率)

∫dx/1+x^2 =arctanx
lim(x→+∞)arctanx=π/2
lim(x→-∞)arctanx=-π/2
所以原式=π/2-(-π/2)=π