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人気のある微分積分 >

integral of 4x(3x-1)^5

解

\int 4 x (3 x - 1)^{5} d x

解

\frac{2}{189} (6 (3 x - 1)^{7} + 7 (3 x - 1)^{6}) + C

解答ステップ

\int 4 x (3 x - 1)^{5} d x

定数を除く:

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

= 4 \cdot \int x (3 x - 1)^{5} d x

u置換積分法を適用する

= 4 \cdot \int \frac{u ^{5} ( u + 1 )}{9} d u

定数を除く:

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

= 4 \cdot \frac{1}{9} \cdot \int u^{5} (u + 1) d u

拡張

u^{5} (u + 1) : u^{6} + u^{5}

= 4 \cdot \frac{1}{9} \cdot \int u^{6} + u^{5} 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

= 4 \cdot \frac{1}{9} (\int u^{6} d u + \int u^{5} d u)

\int u^{6} d u = \frac{u ^{7}}{7}

\int u^{5} d u = \frac{u ^{6}}{6}

= 4 \cdot \frac{1}{9} (\frac{u ^{7}}{7} + \frac{u ^{6}}{6})

代用を戻す

u = 3 x - 1

= 4 \cdot \frac{1}{9} (\frac{( 3 x - 1 ) ^{7}}{7} + \frac{( 3 x - 1 ) ^{6}}{6})

簡素化

4 \cdot \frac{1}{9} (\frac{( 3 x - 1 ) ^{7}}{7} + \frac{( 3 x - 1 ) ^{6}}{6}) : \frac{2}{189} (6 (3 x - 1)^{7} + 7 (3 x - 1)^{6})

= \frac{2}{189} (6 (3 x - 1)^{7} + 7 (3 x - 1)^{6})

定数を解答に追加する

= \frac{2}{189} (6 (3 x - 1)^{7} + 7 (3 x - 1)^{6}) + C

グラフ

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