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受欢迎的微积分 >

integral of e^{(-y^2)/2}

解答

\int e^{\frac{- y ^{2}}{2}} d y

解答

(\frac{π}{2})^{\frac{1}{2}} erf (\frac{y}{2}) + C

求解步骤

\int e^{\frac{- y ^{2}}{2}} d y

化简

= \int e^{- \frac{y ^{2}}{2}} d y

使用指数法则:

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

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

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

使用换元积分法

= \int 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 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)

u = \frac{y}{2}

代回

= 2 \frac{π}{2} erf (\frac{y}{2})

化简

2 \frac{π}{2} erf (\frac{y}{2}) : (\frac{π}{2})^{\frac{1}{2}} erf (\frac{y}{2})

= (\frac{π}{2})^{\frac{1}{2}} erf (\frac{y}{2})

解答补常数

= (\frac{π}{2})^{\frac{1}{2}} erf (\frac{y}{2}) + C

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