integral from 0 to 1 of 1/(5x-10)

解答

\int_{0}^{1} \frac{1}{5 x - 10} d x

- \frac{1}{5} ln (2)

十进制

- 0.13862 \dots

求解步骤

\int_{0}^{1} \frac{1}{5 x - 10} d x

使用换元积分法

= \int_{- 10}^{- 5} \frac{1}{5 u} d u

提出常数:

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

= \frac{1}{5} \cdot \int_{- 10}^{- 5} \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{5} [ln ∣ u ∣]_{- 10}^{- 5}

带入上下限计算后得:

ln (5) - ln (10)

= \frac{1}{5} (ln (5) - ln (10))

化简

= - \frac{1}{5} ln (2)

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