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人気のある微分積分 >

integral from 1 to e of (ln(x))^7

解

\int_{1}^{e} (ln (x))^{7} d x

解

- 1854 e + 5040

+1

十進法表記

0.30549 \dots

解答ステップ

\int_{1}^{e} (ln (x))^{7} d x

簡素化

= \int_{1}^{e} ln^{7} (x) d x

u置換積分法を適用する

= \int_{0}^{1} u^{7} e^{u} d u

部分積分を適用する

= [u^{7} e^{u} - \int 7 u^{6} e^{u} d u]_{0}^{1}

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

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

限度を計算する:

- 1854 e + 5040

= - 1854 e + 5040

グラフ

人気の例

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

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\int_{0}^{6} 4 + 2 x d x

integral from 0 to 1 of (1-x^4)^3

\int_{0}^{1} (1 - x^{4})^{3} d x

integral from 2 to infinity of ln(2x-3)

\int_{2}^{\infty} ln (2 x - 3) d x

integral from 0 to 1 of (1-x^2)^7

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