integral from 0 to 1/9 of arccos(9x)

解

\int_{0}^{\frac{1}{9}} arccos (9 x) d x

\frac{1}{9}

十進法表記

0.11111 \dots

解答ステップ

\int_{0}^{\frac{1}{9}} arccos (9 x) d x

部分積分を適用する

= [x arccos (9 x) - \int - \frac{9 x}{1 - 81 x ^{2}} d x]_{0}^{\frac{1}{9}}

\int - \frac{9 x}{1 - 81 x ^{2}} d x = \frac{1}{9} 1 - 81 x^{2}

= [x arccos (9 x) - \frac{1}{9} 1 - 81 x^{2}]_{0}^{\frac{1}{9}}

簡素化

[x arccos (9 x) - \frac{1}{9} 1 - 81 x^{2}]_{0}^{\frac{1}{9}} : [\frac{1}{9} (9 x arccos (9 x) - 1 - 81 x^{2})]_{0}^{\frac{1}{9}}

= [\frac{1}{9} (9 x arccos (9 x) - 1 - 81 x^{2})]_{0}^{\frac{1}{9}}

限度を計算する:

\frac{1}{9}

= \frac{1}{9}