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^{1/2 x^2}

解答

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

解答

\frac{π}{2} erfi (\frac{x}{2}) + C

求解步骤

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

化简

\frac{1}{2} x^{2} : \frac{x ^{2}}{2}

= \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} erfi (u)

= 2 \frac{π}{2} erfi (u)

u = \frac{x}{2}

代回

= 2 \frac{π}{2} erfi (\frac{x}{2})

化简

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

= \frac{π}{2} erfi (\frac{x}{2})

解答补常数

= \frac{π}{2} erfi (\frac{x}{2}) + C

流行的例子

limit as x approaches 4-of |x-4|

y^'=y+x^2e^x+1-1/e

(\partial)/(\partial x)(3xy^2+2ycos(x))

integral of 1/(x+xsqrt(x))

integral of 8x+1/(sqrt(x))