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人気のある微分積分 >

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

解

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

解

\frac{2 e ^{3} + 10}{9}

+1

十進法表記

5.57456 \dots

解答ステップ

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

部分積分を適用する

= [ln (x) (x + \frac{x ^{3}}{3}) - \int \frac{1}{3} (3 + x^{2}) d x]_{1}^{e}

\int \frac{1}{3} (3 + x^{2}) d x = x + \frac{x ^{3}}{9}

= [ln (x) (x + \frac{x ^{3}}{3}) - (x + \frac{x ^{3}}{9})]_{1}^{e}

簡素化

[ln (x) (x + \frac{x ^{3}}{3}) - (x + \frac{x ^{3}}{9})]_{1}^{e} : [\frac{1}{9} (9 x ln (x) + 3 x^{3} ln (x) - 9 x - x^{3})]_{1}^{e}

= [\frac{1}{9} (9 x ln (x) + 3 x^{3} ln (x) - 9 x - x^{3})]_{1}^{e}

限度を計算する:

\frac{2 e ^{3}}{9} + \frac{10}{9}

= \frac{2 e ^{3}}{9} + \frac{10}{9}

簡素化

= \frac{2 e ^{3} + 10}{9}

グラフ

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