integral from 1 to e of (ln(x))^5

解答

\int_{1}^{e} (ln (x))^{5} d x

- 44 e + 120

十进制

0.39559 \dots

求解步骤

\int_{1}^{e} (ln (x))^{5} d x

化简

= \int_{1}^{e} ln^{5} (x) d x

使用分布积分法

= [x ln^{5} (x) - \int 5 ln^{4} (x) d x]_{1}^{e}

\int 5 ln^{4} (x) d x = 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))

= [x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))]_{1}^{e}

带入上下限计算后得:

- 44 e + 120

= - 44 e + 120

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