integral of (2-x^2)^2

解

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

4 x - \frac{4 x ^{3}}{3} + \frac{x ^{5}}{5} + C

解答ステップ

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

拡張

(2 - x^{2})^{2} : 4 - 4 x^{2} + x^{4}

= \int 4 - 4 x^{2} + x^{4} 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 4 d x - \int 4 x^{2} d x + \int x^{4} d x

\int 4 d x = 4 x

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int x^{4} d x = \frac{x ^{5}}{5}

= 4 x - \frac{4 x ^{3}}{3} + \frac{x ^{5}}{5}

定数を解答に追加する

= 4 x - \frac{4 x ^{3}}{3} + \frac{x ^{5}}{5} + C