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人気のある微分積分 >

integral from 1 to e of (ln(x))^8

解

\int_{1}^{e} (ln (x))^{8} d x

解

14833 e - 40320

+1

十進法表記

0.27436 \dots

解答ステップ

\int_{1}^{e} (ln (x))^{8} d x

簡素化

= \int_{1}^{e} ln^{8} (x) d x

u置換積分法を適用する

= \int_{0}^{1} u^{8} e^{u} d u

部分積分を適用する

= [u^{8} e^{u} - \int 8 u^{7} e^{u} d u]_{0}^{1}

\int 8 u^{7} e^{u} d u = 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u})))))))

= [u^{8} e^{u} - 8 (e^{u} u^{7} - 7 (e^{u} u^{6} - 6 (e^{u} u^{5} - 5 (e^{u} u^{4} - 4 (e^{u} u^{3} - 3 (e^{u} u^{2} - 2 (e^{u} u - e^{u})))))))]_{0}^{1}

限度を計算する:

14833 e - 40320

= 14833 e - 40320

グラフ

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