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

解答

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

\frac{1}{3}

十进制

0.33333 \dots

求解步骤

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

化简

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

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

使用指数法则:

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

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

使用分布积分法

= [- \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}

化简

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

带入上下限计算后得:

\frac{1}{3}

= \frac{1}{3}

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