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受欢迎的微积分 >

integral from 0 to 10 of 1/4 e^{-x/4}

解答

\int_{0}^{10} \frac{1}{4} e^{- \frac{x}{4}} d x

解答

1 - \frac{1}{e ^{\frac{5}{2}}}

+1

十进制

0.91791 \dots

求解步骤

\int_{0}^{10} \frac{1}{4} e^{- \frac{x}{4}} d x

提出常数:

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

= \frac{1}{4} \cdot \int_{0}^{10} e^{- \frac{x}{4}} d x

使用换元积分法

= \frac{1}{4} \cdot \int_{0}^{- \frac{5}{2}} - 4 e^{u} d u

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

= \frac{1}{4} (- \int_{- \frac{5}{2}}^{0} - 4 e^{u} d u)

提出常数:

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

= \frac{1}{4} (- (- 4 \cdot \int_{- \frac{5}{2}}^{0} e^{u} d u))

使用积分常见定则:

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

= \frac{1}{4} (- (- 4 [e^{u}]_{- \frac{5}{2}}^{0}))

化简

\frac{1}{4} (- (- 4 [e^{u}]_{- \frac{5}{2}}^{0})) : [e^{u}]_{- \frac{5}{2}}^{0}

= [e^{u}]_{- \frac{5}{2}}^{0}

带入上下限计算后得:

1 - \frac{1}{e ^{\frac{5}{2}}}

= 1 - \frac{1}{e ^{\frac{5}{2}}}

作图

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