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受欢迎的微积分 >

integral of 1/(sqrt(2pi))e^{(x^2)/2}

解答

\int \frac{1}{2 π} e^{\frac{x ^{2}}{2}} d x

解答

\frac{1}{2} erfi (\frac{x}{2}) + C

求解步骤

\int \frac{1}{2 π} e^{\frac{x ^{2}}{2}} d x

提出常数:

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

= \frac{1}{2 π} \cdot \int e^{\frac{x ^{2}}{2}} d x

2 π = 2^{\frac{1}{2}} π^{\frac{1}{2}}

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} \cdot \int 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}

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} \cdot \int e^{(\frac{x}{2})^{2}} d x

使用换元积分法

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} \cdot \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

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} 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} erfi (u)

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} 2 \frac{π}{2} erfi (u)

u = \frac{x}{2}

代回

= \frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} 2 \frac{π}{2} erfi (\frac{x}{2})

化简

\frac{1}{2 ^{\frac{1}{2}} π ^{\frac{1}{2}}} 2 \frac{π}{2} erfi (\frac{x}{2}) : \frac{1}{2} erfi (\frac{x}{2})

= \frac{1}{2} erfi (\frac{x}{2})

解答补常数

= \frac{1}{2} erfi (\frac{x}{2}) + C

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