integral from 2 to 8 of (e^{1/x})/(x^2)

解答

\int_{2}^{8} \frac{e ^{\frac{1}{x}}}{x ^{2}} d x

e - 8 e

十进制

0.51557 \dots

求解步骤

\int_{2}^{8} \frac{e ^{\frac{1}{x}}}{x ^{2}} d x

使用换元积分法

= \int_{\frac{1}{2}}^{\frac{1}{8}} - e^{u} d u

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

= - \int_{\frac{1}{8}}^{\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

= - (- \int_{\frac{1}{8}}^{\frac{1}{2}} e^{u} d u)

使用积分常见定则:

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

= - (- [e^{u}]_{\frac{1}{8}}^{\frac{1}{2}})

化简

= [e^{u}]_{\frac{1}{8}}^{\frac{1}{2}}

带入上下限计算后得:

e - 8 e

= e - 8 e

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

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