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 t^2}

解答

\int e^{- \frac{1}{2} t^{2}} d t

解答

\frac{π}{2} erf (\frac{t}{2}) + C

求解步骤

\int e^{- \frac{1}{2} t^{2}} d t

化简

- \frac{1}{2} t^{2} : - \frac{t ^{2}}{2}

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

使用指数法则:

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

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

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

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{t}{2}

代回

= 2 \frac{π}{2} erf (\frac{t}{2})

化简

2 \frac{π}{2} erf (\frac{t}{2}) : \frac{π}{2} erf (\frac{t}{2})

= \frac{π}{2} erf (\frac{t}{2})

解答补常数

= \frac{π}{2} erf (\frac{t}{2}) + C

流行的例子

integral of sin(2θ)sin(4θ)

\int sin (2 θ) sin (4 θ) d θ

integral of (x^{2.7}+x^{-1.3})

\int (x^{2.7} + x^{- 1.3}) d x

integral of (7x^3-5x^2+7x-8)/(x^2)

\int \frac{7 x ^{3} - 5 x ^{2} + 7 x - 8}{x ^{2}} d x

integral of (x^3)/(x^2-x-6)

\int \frac{x ^{3}}{x ^{2} - x - 6} d x

integral of (7x^2-x-5)/(x^2-1)

\int \frac{7 x ^{2} - x - 5}{x ^{2} - 1} d x