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人気のある微分積分 >

integral of x(1-x)^{2/3}

解

\int x (1 - x)^{\frac{2}{3}} d x

解

\frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}} + C

解答ステップ

\int x (1 - x)^{\frac{2}{3}} d x

u置換積分法を適用する

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

定数を除く:

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

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

拡張

(- u + 1) u^{\frac{2}{3}} : - u^{\frac{5}{3}} + u^{\frac{2}{3}}

= - \int - u^{\frac{5}{3}} + u^{\frac{2}{3}} 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

= - (- \int u^{\frac{5}{3}} d u + \int u^{\frac{2}{3}} d u)

\int u^{\frac{5}{3}} d u = \frac{3}{8} u^{\frac{8}{3}}

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

= - (- \frac{3}{8} u^{\frac{8}{3}} + \frac{3}{5} u^{\frac{5}{3}})

代用を戻す

u = 1 - x

= - (- \frac{3}{8} (1 - x)^{\frac{8}{3}} + \frac{3}{5} (1 - x)^{\frac{5}{3}})

簡素化

= \frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}}

定数を解答に追加する

= \frac{3}{8} (1 - x)^{\frac{8}{3}} - \frac{3}{5} (1 - x)^{\frac{5}{3}} + C

グラフ

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