integral from 0 to 1 of x(1-x)^8

解

\int_{0}^{1} x (1 - x)^{8} d x

\frac{1}{90}

十進法表記

0.01111 \dots

解答ステップ

\int_{0}^{1} x (1 - x)^{8} d x

部分積分を適用する

= [- \frac{1}{9} x (1 - x)^{9} - \int - \frac{1}{9} (1 - x)^{9} d x]_{0}^{1}

\int - \frac{1}{9} (1 - x)^{9} d x = \frac{1}{90} (1 - x)^{10}

= [- \frac{1}{9} x (1 - x)^{9} - \frac{1}{90} (1 - x)^{10}]_{0}^{1}

限度を計算する:

\frac{1}{90}

= \frac{1}{90}