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人気のある微分積分 >

integral of (x^3)/(\sqrt[3]{3-2x^2)}

解

\int \frac{x ^{3}}{3 3 - 2 x ^{2}} d x

解

\frac{1}{8} (\frac{3}{5} (3 - 2 x^{2})^{\frac{5}{3}} - \frac{9}{2} (3 - 2 x^{2})^{\frac{2}{3}}) + C

解答ステップ

\int \frac{x ^{3}}{3 3 - 2 x ^{2}} d x

u置換積分法を適用する

= \int \frac{u - 3}{8 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}{8} \cdot \int \frac{u - 3}{3 u} d u

拡張

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

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

\int u^{\frac{2}{3}} d u = \frac{3}{5} u^{\frac{5}{3}}

\int \frac{3}{3 u} d u = \frac{9}{2} u^{\frac{2}{3}}

= \frac{1}{8} (\frac{3}{5} u^{\frac{5}{3}} - \frac{9}{2} u^{\frac{2}{3}})

代用を戻す

u = 3 - 2 x^{2}

= \frac{1}{8} (\frac{3}{5} (3 - 2 x^{2})^{\frac{5}{3}} - \frac{9}{2} (3 - 2 x^{2})^{\frac{2}{3}})

定数を解答に追加する

= \frac{1}{8} (\frac{3}{5} (3 - 2 x^{2})^{\frac{5}{3}} - \frac{9}{2} (3 - 2 x^{2})^{\frac{2}{3}}) + C

グラフ

人気の例

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x \to - 2 lim (\frac{3 x - 9}{x + 1})

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n = 2 \sum \infty \frac{1}{( - 5 ) ^{n}}

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integral of (x^{15})/(sqrt(x^{16)+1)}

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