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受欢迎的微积分 >

integral from 0 to 10 of (5e^{-0.05x})

解答

\int_{0}^{10} (5 e^{- 0.05 x}) d x

解答

39.34693 \dots

求解步骤

\int_{0}^{10} 5 e^{- 0.05 x} d x

提出常数:

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

= 5 \cdot \int_{0}^{10} e^{- 0.05 x} d x

使用换元积分法

= 5 \cdot \int_{0}^{- 0.5} - \frac{1}{0.05} e^{u} d u

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

= 5 (- \int_{- 0.5}^{0} - \frac{1}{0.05} e^{u} d u)

提出常数:

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

= 5 (- (- \frac{1}{0.05} \cdot \int_{- 0.5}^{0} e^{u} d u))

化简

= 5 (- (- 20 \cdot \int_{- 0.5}^{0} e^{u} d u))

使用积分常见定则:

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

= 5 (- (- 20 [e^{u}]_{- 0.5}^{0}))

化简

= 100 [e^{u}]_{- 0.5}^{0}

带入上下限计算后得:

0.39346 \dots

= 100 \cdot 0.39346 \dots

化简

= 39.34693 \dots

作图

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