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受欢迎的微积分 >

integral from 0 to 1 of 3e^{-2x}

解答

\int_{0}^{1} 3 e^{- 2 x} d x

解答

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

+1

十进制

1.29699 \dots

求解步骤

\int_{0}^{1} 3 e^{- 2 x} d x

提出常数:

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

= 3 \cdot \int_{0}^{1} e^{- 2 x} d x

使用换元积分法

= 3 \cdot \int_{0}^{- 2} - \frac{1}{2} e^{u} d u

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

= 3 (- \int_{- 2}^{0} - \frac{1}{2} e^{u} d u)

提出常数:

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

= 3 (- (- \frac{1}{2} \cdot \int_{- 2}^{0} e^{u} d u))

使用积分常见定则:

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

= 3 (- (- \frac{1}{2} [e^{u}]_{- 2}^{0}))

化简

3 (- (- \frac{1}{2} [e^{u}]_{- 2}^{0})) : \frac{3}{2} [e^{u}]_{- 2}^{0}

= \frac{3}{2} [e^{u}]_{- 2}^{0}

带入上下限计算后得:

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

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

化简

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

作图

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