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人気のある微分積分 >

integral from 1 to 2 of (x-2/x)^2

解

\int_{1}^{2} (x - \frac{2}{x})^{2} d x

解

\frac{1}{3}

+1

十進法表記

0.33333 \dots

解答ステップ

\int_{1}^{2} (x - \frac{2}{x})^{2} d x

簡素化

x - \frac{2}{x} : \frac{x ^{2} - 2}{x}

= \int_{1}^{2} (\frac{x ^{2} - 2}{x})^{2} d x

指数の規則を適用する:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \int_{1}^{2} \frac{( x ^{2} - 2 ) ^{2}}{x ^{2}} d x

部分積分を適用する

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

\int - 4 (x^{2} - 2) d x = - 4 (\frac{x ^{3}}{3} - 2 x)

= [- \frac{( x ^{2} - 2 ) ^{2}}{x} - (- 4 (\frac{x ^{3}}{3} - 2 x))]_{1}^{2}

簡素化

= [- \frac{( x ^{2} - 2 ) ^{2}}{x} + 4 (\frac{x ^{3}}{3} - 2 x)]_{1}^{2}

限度を計算する:

\frac{1}{3}

= \frac{1}{3}

グラフ

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