返回题库

高斯积分：$\int_0^{\infty}e^{-x^2

Multiple Integrals

专题
General / 综合
难度
L4

题目详情

计算 ∫0∞e−x2/2 dx\displaystyle\int_0^{\infty} e^{-x^2/2}\,dx。

Evaluate ∫0∞e− x2/2 dx\int_0^\infty e^{-\,x^2/2}\,dx?

解析

经典高斯积分：把一维积分平方并转到二维极坐标，可得

∫0∞e−x2/2 dx=π2.\int_0^{\infty} e^{-x^2/2}\,dx=\sqrt{\frac{\pi}{2}}.

Original Explanation

By extending to a two-dimensional integral and using polar coordinates, one obtains ∫0∞e− x2/2 dx=π2.\int_0^\infty e^{-\,x^2/2}\,dx = \sqrt{\frac{\pi}{2}}.