integral from 0 to 1 of e^{8x-7}

解答

\int_{0}^{1} e^{8 x - 7} d x

\frac{e ^{8} - 1}{8 e ^{7}}

十进制

0.33967 \dots

求解步骤

\int_{0}^{1} e^{8 x - 7} d x

使用换元积分法

= \int_{- 7}^{1} e^{u} \frac{1}{8} d u

提出常数:

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

= \frac{1}{8} \cdot \int_{- 7}^{1} e^{u} d u

使用积分常见定则:

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

= \frac{1}{8} [e^{u}]_{- 7}^{1}

带入上下限计算后得:

e - \frac{1}{e ^{7}}

= \frac{1}{8} (e - \frac{1}{e ^{7}})

化简

= \frac{e ^{8} - 1}{8 e ^{7}}

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