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人気のある微分積分 >

integral of (1372x)\sqrt[3]{7x+1}

解

\int (1372 x) 3 7 x + 1 d x

解

28 (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}}) + C

解答ステップ

\int 1372 x 3 7 x + 1 d x

定数を除く:

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

= 1372 \cdot \int x 3 7 x + 1 d x

u置換積分法を適用する

= 1372 \cdot \int \frac{3 u ( u - 1 )}{49} d u

定数を除く:

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

= 1372 \cdot \frac{1}{49} \cdot \int 3 u (u - 1) d u

拡張

3 u (u - 1) : u^{\frac{4}{3}} - 3 u

= 1372 \cdot \frac{1}{49} \cdot \int u^{\frac{4}{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

= 1372 \cdot \frac{1}{49} (\int u^{\frac{4}{3}} d u - \int 3 u d u)

\int u^{\frac{4}{3}} d u = \frac{3}{7} u^{\frac{7}{3}}

\int 3 u d u = \frac{3}{4} u^{\frac{4}{3}}

= 1372 \cdot \frac{1}{49} (\frac{3}{7} u^{\frac{7}{3}} - \frac{3}{4} u^{\frac{4}{3}})

代用を戻す

u = 7 x + 1

= 1372 \cdot \frac{1}{49} (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}})

簡素化

1372 \cdot \frac{1}{49} (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}}) : 28 (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}})

= 28 (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}})

定数を解答に追加する

= 28 (\frac{3}{7} (7 x + 1)^{\frac{7}{3}} - \frac{3}{4} (7 x + 1)^{\frac{4}{3}}) + C

グラフ

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