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受欢迎的微积分 >

integral from 0 to 1 of e^{1-3t}

解答

\int_{0}^{1} e^{1 - 3 t} d t

解答

\frac{e ^{3} - 1}{3 e ^{2}}

+1

十进制

0.86098 \dots

求解步骤

\int_{0}^{1} e^{1 - 3 t} d t

使用换元积分法

= \int_{1}^{- 2} - \frac{1}{3} e^{u} d u

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

= - \int_{- 2}^{1} - \frac{1}{3} e^{u} d u

提出常数:

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

= - (- \frac{1}{3} \cdot \int_{- 2}^{1} e^{u} d u)

使用积分常见定则:

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

= - (- \frac{1}{3} [e^{u}]_{- 2}^{1})

化简

= \frac{1}{3} [e^{u}]_{- 2}^{1}

带入上下限计算后得:

e - \frac{1}{e ^{2}}

= \frac{1}{3} (e - \frac{1}{e ^{2}})

化简

= \frac{e ^{3} - 1}{3 e ^{2}}

作图

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