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人気のある微分積分 >

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

解

\int \frac{x ^{3}}{3 x ^{2} + 7} d x

解

\frac{1}{20} (6 (x^{2} + 7)^{\frac{5}{3}} - 105 (x^{2} + 7)^{\frac{2}{3}}) + C

解答ステップ

\int \frac{x ^{3}}{3 x ^{2} + 7} d x

u置換積分法を適用する

= \int \frac{u ^{\frac{1}{3}}}{3 3 u ^{\frac{2}{3}} + 7} d u

定数を除く:

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

= \frac{1}{3} \cdot \int \frac{u ^{\frac{1}{3}}}{3 u ^{\frac{2}{3}} + 7} d u

u置換積分法を適用する

= \frac{1}{3} \cdot \int \frac{3 ( v - 7 )}{2 3 v} d v

定数を除く:

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

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int \frac{v - 7}{3 v} d v

拡張

\frac{v - 7}{3 v} : v^{\frac{2}{3}} - \frac{7}{3 v}

= \frac{1}{3} \cdot \frac{3}{2} \cdot \int v^{\frac{2}{3}} - \frac{7}{3 v} d v

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}{3} \cdot \frac{3}{2} (\int v^{\frac{2}{3}} d v - \int \frac{7}{3 v} d v)

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

\int \frac{7}{3 v} d v = \frac{21}{2} v^{\frac{2}{3}}

= \frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} v^{\frac{5}{3}} - \frac{21}{2} v^{\frac{2}{3}})

代用を戻す

= \frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} ((x^{3})^{\frac{2}{3}} + 7)^{\frac{5}{3}} - \frac{21}{2} ((x^{3})^{\frac{2}{3}} + 7)^{\frac{2}{3}})

簡素化

\frac{1}{3} \cdot \frac{3}{2} (\frac{3}{5} ((x^{3})^{\frac{2}{3}} + 7)^{\frac{5}{3}} - \frac{21}{2} ((x^{3})^{\frac{2}{3}} + 7)^{\frac{2}{3}}) : \frac{1}{20} (6 (x^{2} + 7)^{\frac{5}{3}} - 105 (x^{2} + 7)^{\frac{2}{3}})

= \frac{1}{20} (6 (x^{2} + 7)^{\frac{5}{3}} - 105 (x^{2} + 7)^{\frac{2}{3}})

定数を解答に追加する

= \frac{1}{20} (6 (x^{2} + 7)^{\frac{5}{3}} - 105 (x^{2} + 7)^{\frac{2}{3}}) + C

グラフ

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