integral from 0.1 to 0.3 of e^{-2x^2}

解答

\int_{0.1}^{0.3} e^{- 2 x^{2}} d x

0.18359 \dots

求解步骤

\int_{0.1}^{0.3} e^{- 2 x^{2}} d x

= \int_{0.1}^{0.3} e^{- (2 x)^{2}} d x

使用换元积分法

= \int_{0.14142 \dots}^{0.42426 \dots} e^{- u^{2}} \frac{1}{2} d u

提出常数:

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

= \frac{1}{2} \cdot \int_{0.14142 \dots}^{0.42426 \dots} 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)

= \frac{1}{2} [\frac{π}{2} erf (u)]_{0.14142 \dots}^{0.42426 \dots}

化简

= 0.70710 \dots [0.88622 \dots erf (u)]_{0.14142 \dots}^{0.42426 \dots}

带入上下限计算后得:

0.25964 \dots

= 0.70710 \dots \cdot 0.25964 \dots

化简

= 0.18359 \dots

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