integral from 0 to l of (x/l)^2

解答

\int_{0}^{l} (\frac{x}{l})^{2} d x

\frac{l}{3}

求解步骤

\int_{0}^{l} (\frac{x}{l})^{2} d x

使用分布积分法

= [\frac{x ^{3}}{l ^{2}} - \int \frac{2 x ^{2}}{l ^{2}} d x]_{0}^{l}

\int \frac{2 x ^{2}}{l ^{2}} d x = \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3}

= [\frac{x ^{3}}{l ^{2}} - \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3}]_{0}^{l}

化简

\frac{x ^{3}}{l ^{2}} - \frac{2}{l ^{2}} \cdot \frac{x ^{3}}{3} : \frac{x ^{3}}{3 l ^{2}}

= [\frac{x ^{3}}{3 l ^{2}}]_{0}^{l}

带入上下限计算后得:

\frac{l}{3}

= \frac{l}{3}

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