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 0 to r of 0.5e^{-0.5x}

解答

\int_{0}^{r} 0.5 e^{- 0.5 x} d x

解答

- e^{- 0.5 r} + 1

求解步骤

\int_{0}^{r} 0.5 e^{- 0.5 x} d x

提出常数:

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

= 0.5 \cdot \int_{0}^{r} e^{- 0.5 x} d x

使用换元积分法

= 0.5 \cdot \int_{0}^{- 0.5 r} - \frac{1}{0.5} e^{u} d u

提出常数:

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

= 0.5 (- \frac{1}{0.5} \cdot \int_{0}^{- 0.5 r} e^{u} d u)

化简

= 0.5 (- 2 \cdot \int_{0}^{- 0.5 r} e^{u} d u)

使用积分常见定则:

\int e^{u} d u = e^{u}

= 0.5 (- 2 [e^{u}]_{0}^{- 0.5 r})

化简

= - [e^{u}]_{0}^{- 0.5 r}

带入上下限计算后得:

e^{- 0.5 r} - 1

= - (e^{- 0.5 r} - 1)

化简

= - e^{- 0.5 r} + 1

流行的例子

integral from 1 to 2 of x^6ln(x)

\int_{1}^{2} x^{6} ln (x) d x

integral from 0 to pi of cos(x)-0.5

\int_{0}^{π} cos (x) - 0.5 d x

integral from 0 to 2 of 5e^x

\int_{0}^{2} 5 e^{x} d x

integral from-3 to 9 of-xcos(x)

\int_{- 3}^{9} - x cos (x) d x

integral from 50 to 100 of x/(27+4x^2)

\int_{50}^{100} \frac{x}{27 + 4 x ^{2}} d x