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人気のある微分積分 >

integral from 1 to e of x^4(ln(x))^2

解

\int_{1}^{e} x^{4} (ln (x))^{2} d x

解

\frac{17 e ^{5} - 2}{125}

+1

十進法表記

20.16818 \dots

解答ステップ

\int_{1}^{e} x^{4} (ln (x))^{2} d x

部分積分を適用する

= [\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \int \frac{2 x ^{4} ln ( x )}{5} d x)]_{1}^{e}

\int \frac{2 x ^{4} ln ( x )}{5} d x = \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25})

= [\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

簡素化

[\frac{1}{5} (x^{5} ln^{2} (x) - 5 \cdot \frac{2}{5} (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e} : [\frac{1}{5} (x^{5} ln^{2} (x) - 2 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

= [\frac{1}{5} (x^{5} ln^{2} (x) - 2 (\frac{1}{5} x^{5} ln (x) - \frac{x ^{5}}{25}))]_{1}^{e}

限度を計算する:

\frac{17 e ^{5}}{125} - \frac{2}{125}

= \frac{17 e ^{5}}{125} - \frac{2}{125}

簡素化

= \frac{17 e ^{5} - 2}{125}

グラフ

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