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人気のある微分積分 >

integral of x(x+3)^7

解

\int x (x + 3)^{7} d x

解

\frac{1}{72} (x + 3)^{8} (8 x - 3) + C

解答ステップ

\int x (x + 3)^{7} d x

u置換積分法を適用する

= \int \frac{7 u ( 7 u - 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}{7} \cdot \int 7 u (7 u - 3) d u

拡張

7 u (7 u - 3) : u^{\frac{2}{7}} - 3 7 u

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

\int u^{\frac{2}{7}} d u = \frac{7}{9} u^{\frac{9}{7}}

\int 3 7 u d u = \frac{21}{8} u^{\frac{8}{7}}

= \frac{1}{7} (\frac{7}{9} u^{\frac{9}{7}} - \frac{21}{8} u^{\frac{8}{7}})

代用を戻す

u = (x + 3)^{7}

= \frac{1}{7} (\frac{7}{9} ((x + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((x + 3)^{7})^{\frac{8}{7}})

簡素化

\frac{1}{7} (\frac{7}{9} ((x + 3)^{7})^{\frac{9}{7}} - \frac{21}{8} ((x + 3)^{7})^{\frac{8}{7}}) : \frac{1}{72} (x + 3)^{8} (8 x - 3)

= \frac{1}{72} (x + 3)^{8} (8 x - 3)

定数を解答に追加する

= \frac{1}{72} (x + 3)^{8} (8 x - 3) + C

グラフ

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