integral from 1 to e of (ln(x))^9

解

\int_{1}^{e} (ln (x))^{9} d x

- 133496 e + 362880

十進法表記

0.24902 \dots

解答ステップ

\int_{1}^{e} (ln (x))^{9} d x

簡素化

= \int_{1}^{e} ln^{9} (x) d x

部分積分を適用する

= [x ln^{9} (x) - \int 9 ln^{8} (x) d x]_{1}^{e}

\int 9 ln^{8} (x) d x = 9 (x ln^{8} (x) - 8 (x ln^{7} (x) - 7 (x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))))))

= [x ln^{9} (x) - 9 (x ln^{8} (x) - 8 (x ln^{7} (x) - 7 (x ln^{6} (x) - 6 (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}

限度を計算する:

- 133496 e + 362880

= - 133496 e + 362880