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受欢迎的微积分 >

integral from 0 to 1 of e^{-1/2 x^2}

解答

\int_{0}^{1} e^{- \frac{1}{2} x^{2}} d x

解答

0.85562 \dots

求解步骤

\int_{0}^{1} e^{- \frac{1}{2} x^{2}} d x

化简

- \frac{1}{2} x^{2} : - \frac{x ^{2}}{2}

= \int_{0}^{1} e^{- \frac{x ^{2}}{2}} d x

使用指数法则:

\frac{a ^{n}}{b ^{n}} = (\frac{a}{b})^{n}

\frac{x ^{2}}{2} = (\frac{x}{2})^{2}

= \int_{0}^{1} e^{- (\frac{x}{2})^{2}} d x

使用换元积分法

= \int_{0}^{\frac{1}{2}} e^{- u^{2}} 2 d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{0}^{\frac{1}{2}} e^{- u^{2}} d u

这是非初等函数积分:

\int e^{- u^{2}} d u = \frac{π}{2} erf (u)

= 2 [\frac{π}{2} erf (u)]_{0}^{\frac{1}{2}}

带入上下限计算后得:

0.60501 \dots

= 2 \cdot 0.60501 \dots

化简

= 0.85562 \dots

作图

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