integral of e^{(-x^2)/a}

解答

\int e^{\frac{- x ^{2}}{a}} d x

a \frac{π}{2} erf (\frac{x}{a}) + C

求解步骤

\int e^{\frac{- x ^{2}}{a}} d x

化简

= \int e^{- \frac{x ^{2}}{a}} d x

使用指数法则:

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

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

= \int e^{- (\frac{x}{a})^{2}} d x

使用换元积分法

= \int e^{- u^{2}} a d u

提出常数:

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

= a \cdot \int 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)

= a \frac{π}{2} erf (u)

u = \frac{x}{a}

代回

= a \frac{π}{2} erf (\frac{x}{a})

解答补常数

= a \frac{π}{2} erf (\frac{x}{a}) + C

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