integral from-1 to 1 of (arccos(x))^2

解

\int_{- 1}^{1} (arccos (x))^{2} d x

π^{2} - 4

十進法表記

5.86960 \dots

解答ステップ

\int_{- 1}^{1} (arccos (x))^{2} d x

簡素化

= \int_{- 1}^{1} arccos^{2} (x) d x

部分積分を適用する

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

\int - \frac{2 x arccos ( x )}{1 - x ^{2}} d x = 2 (arccos (x) 1 - x^{2} + x)

= [x arccos^{2} (x) - 2 (arccos (x) 1 - x^{2} + x)]_{- 1}^{1}

限度を計算する:

π^{2} - 4

= π^{2} - 4