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 1 of 1/(e^{5t)}

解答

\int_{- 1}^{1} \frac{1}{e ^{5 t}} d t

解答

\frac{e ^{10} - 1}{5 e ^{5}}

+1

十进制

29.68128 \dots

求解步骤

\int_{- 1}^{1} \frac{1}{e ^{5 t}} d t

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{e ^{5 t}} = e^{- 5 t}

= \int_{- 1}^{1} e^{- 5 t} d t

使用换元积分法

= \int_{5}^{- 5} - \frac{1}{5} e^{u} d u

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

= - \int_{- 5}^{5} - \frac{1}{5} e^{u} d u

提出常数:

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

= - (- \frac{1}{5} \cdot \int_{- 5}^{5} e^{u} d u)

使用积分常见定则:

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

= - (- \frac{1}{5} [e^{u}]_{- 5}^{5})

化简

= \frac{1}{5} [e^{u}]_{- 5}^{5}

带入上下限计算后得:

e^{5} - \frac{1}{e ^{5}}

= \frac{1}{5} (e^{5} - \frac{1}{e ^{5}})

化简

= \frac{e ^{10} - 1}{5 e ^{5}}

作图

流行的例子

integral from 1 to 3 of 7x^2

\int_{1}^{3} 7 x^{2} d x

integral from 2 to 5 of (x^3+2)

\int_{2}^{5} (x^{3} + 2) d x

integral from 0 to 0.5 of 1/(1+x^4)

\int_{0}^{0.5} \frac{1}{1 + x ^{4}} d x

integral from 0 to 3 of e^{-y^2}

\int_{0}^{3} e^{- y^{2}} d y

integral from 0 to 3 of x(x-3)(x-1)

\int_{0}^{3} x (x - 3) (x - 1) d x