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人気のある微分積分 >

integral from 0 to sqrt(7 of)(x^3)/(\sqrt[3]{1+x^2)}

解

\int_{0}^{7} \frac{x ^{3}}{3 1 + x ^{2}} d x

解

\frac{141}{20}

+1

十進法表記

7.05

解答ステップ

\int_{0}^{7} \frac{x ^{3}}{3 1 + x ^{2}} d x

u置換積分法を適用する

= \int_{1}^{8} \frac{u - 1}{2 3 u} d u

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int_{1}^{8} \frac{u - 1}{3 u} d u

拡張

\frac{u - 1}{3 u} : u^{\frac{2}{3}} - \frac{1}{3 u}

= \frac{1}{2} \cdot \int_{1}^{8} u^{\frac{2}{3}} - \frac{1}{3 u} d u

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int_{1}^{8} u^{\frac{2}{3}} d u - \int_{1}^{8} \frac{1}{3 u} d u)

\int_{1}^{8} u^{\frac{2}{3}} d u = \frac{93}{5}

\int_{1}^{8} \frac{1}{3 u} d u = \frac{9}{2}

= \frac{1}{2} (\frac{93}{5} - \frac{9}{2})

簡素化

\frac{1}{2} (\frac{93}{5} - \frac{9}{2}) : \frac{141}{20}

= \frac{141}{20}

グラフ

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