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人気のある微分積分 >

integral of x(8x+5)^8

解

\int x (8 x + 5)^{8} d x

解

\frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9}) + C

解答ステップ

\int x (8 x + 5)^{8} d x

u置換積分法を適用する

= \int \frac{u ^{8} ( u - 5 )}{64} d u

定数を除く:

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

= \frac{1}{64} \cdot \int u^{8} (u - 5) d u

拡張

u^{8} (u - 5) : u^{9} - 5 u^{8}

= \frac{1}{64} \cdot \int u^{9} - 5 u^{8} 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}{64} (\int u^{9} d u - \int 5 u^{8} d u)

\int u^{9} d u = \frac{u ^{10}}{10}

\int 5 u^{8} d u = \frac{5 u ^{9}}{9}

= \frac{1}{64} (\frac{u ^{10}}{10} - \frac{5 u ^{9}}{9})

代用を戻す

u = 8 x + 5

= \frac{1}{64} (\frac{( 8 x + 5 ) ^{10}}{10} - \frac{5 ( 8 x + 5 ) ^{9}}{9})

簡素化

\frac{1}{64} (\frac{( 8 x + 5 ) ^{10}}{10} - \frac{5 ( 8 x + 5 ) ^{9}}{9}) : \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9})

= \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9})

定数を解答に追加する

= \frac{1}{5760} (9 (8 x + 5)^{10} - 50 (8 x + 5)^{9}) + C

グラフ

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