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 from 1 to 0 of e^{x^2-x}

解答

\int_{1}^{0} e^{x^{2} - x} d x

解答

- \frac{4 e ( π erfi ( \frac{1}{2} ) - π erfi ( - \frac{1}{2} ) )}{2}

求解步骤

\int_{1}^{0} e^{x^{2} - x} d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{1} e^{x^{2} - x} d x

对

x^{2} - x

配方:

(x - \frac{1}{2})^{2} - \frac{1}{4}

= - \int_{0}^{1} e^{(x - \frac{1}{2})^{2} - \frac{1}{4}} d x

使用换元积分法

= - \int_{- \frac{1}{2}}^{\frac{1}{2}} e^{u^{2} - \frac{1}{4}} d u

化简

e^{u^{2} - \frac{1}{4}} : e^{u^{2}} 4 e

= - \int_{- \frac{1}{2}}^{\frac{1}{2}} e^{u^{2}} 4 e d u

提出常数:

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

= - 4 e \cdot \int_{- \frac{1}{2}}^{\frac{1}{2}} 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)

= - 4 e [\frac{π}{2} erfi (u)]_{- \frac{1}{2}}^{\frac{1}{2}}

带入上下限计算后得:

\frac{π erfi ( \frac{1}{2} ) - π erfi ( - \frac{1}{2} )}{2}

= - 4 e \frac{π erfi ( \frac{1}{2} ) - π erfi ( - \frac{1}{2} )}{2}

化简

= - \frac{4 e ( π erfi ( \frac{1}{2} ) - π erfi ( - \frac{1}{2} ) )}{2}

流行的例子

integral from y to 8-y of 3/(x^2+y^2)

\int_{y}^{8 - y} \frac{3}{x ^{2} + y ^{2}} d y

integral from 1 to 2 of 25(x^2+1)^7

\int_{1}^{2} 25 (x^{2} + 1)^{7} d x

integral from 2 to 3 of 5^{2x}

\int_{2}^{3} 5^{2 x} d x

integral from 0 to 4.3 of \sqrt[3]{3x+4}

\int_{0}^{4.3} 3 3 x + 4 d x

integral from 0 to 1 of sqrt((1-x^2))

\int_{0}^{1} (1 - x^{2}) d x