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

解

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

- \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{x ^{3}}{3} + C

解答ステップ

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

拡張

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

= \int x 1 - x^{2} - 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 x 1 - x^{2} d x - \int x^{2} d x

\int x 1 - x^{2} d x = - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}}

\int x^{2} d x = \frac{x ^{3}}{3}

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{x ^{3}}{3}

定数を解答に追加する

= - \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} - \frac{x ^{3}}{3} + C