integral from 0 to pi of (1-cos(x))^2

解

\int_{0}^{π} (1 - cos (x))^{2} d x

π + \frac{π}{2}

十進法表記

4.71238 \dots

解答ステップ

\int_{0}^{π} (1 - cos (x))^{2} d x

拡張

(1 - cos (x))^{2} : 1 - 2 cos (x) + cos^{2} (x)

= \int_{0}^{π} 1 - 2 cos (x) + cos^{2} (x) d x

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_{0}^{π} 1 d x - \int_{0}^{π} 2 cos (x) d x + \int_{0}^{π} cos^{2} (x) d x

\int_{0}^{π} 1 d x = π

\int_{0}^{π} 2 cos (x) d x = 0

\int_{0}^{π} cos^{2} (x) d x = \frac{π}{2}

= π - 0 + \frac{π}{2}

簡素化

= π + \frac{π}{2}