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人気のある微分積分 >

integral of x^2(1-x)^6

解

\int x^{2} (1 - x)^{6} d x

解

- \frac{1}{7} x^{2} (1 - x)^{7} - \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8}) + C

解答ステップ

\int x^{2} (1 - x)^{6} d x

u置換積分法を適用する

= \int - (- u + 1)^{2} u^{6} d u

定数を除く:

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

= - \int (- u + 1)^{2} u^{6} d u

部分積分を適用する

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

\int - \frac{2}{7} u^{7} (- u + 1) d u = - \frac{1}{252} (- 8 u^{9} + 9 u^{8})

= - (\frac{1}{7} u^{7} (- u + 1)^{2} - (- \frac{1}{252} (- 8 u^{9} + 9 u^{8})))

代用を戻す

u = 1 - x

= - (\frac{1}{7} (1 - x)^{7} (- (1 - x) + 1)^{2} - (- \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8})))

簡素化

- (\frac{1}{7} (1 - x)^{7} (- (1 - x) + 1)^{2} - (- \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8}))) : - \frac{1}{7} x^{2} (1 - x)^{7} - \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8})

= - \frac{1}{7} x^{2} (1 - x)^{7} - \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8})

定数を解答に追加する

= - \frac{1}{7} x^{2} (1 - x)^{7} - \frac{1}{252} (- 8 (1 - x)^{9} + 9 (1 - x)^{8}) + C

グラフ

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