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

解

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

- \frac{π}{2}

十進法表記

- 1.57079 \dots

解答ステップ

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

拡張

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

= \int_{- 1}^{1} x 1 - x^{2} - 1 - x^{2} 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_{- 1}^{1} x 1 - x^{2} d x - \int_{- 1}^{1} 1 - x^{2} d x

\int_{- 1}^{1} x 1 - x^{2} d x = 0

\int_{- 1}^{1} 1 - x^{2} d x = \frac{π}{2}

= 0 - \frac{π}{2}

簡素化

= - \frac{π}{2}