zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

化简

= \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}

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

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{x}{2}

代回

= 2 \frac{π}{2} erf (\frac{x}{2})

化简

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

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

解答补常数

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

流行的例子

integral of 1/((2-y)^3)

integral of (18)/(x^3+x^2)

integral of 7((sin(2x))/(sin(x)))

integral of 1/((sqrt(x)-x))

integral of (xsin(x^2)+tan^2(x))