integral of (9-x^2)^2

解

\int (9 - x^{2})^{2} d x

81 x - 6 x^{3} + \frac{x ^{5}}{5} + C

解答ステップ

\int (9 - x^{2})^{2} d x

拡張

(9 - x^{2})^{2} : 81 - 18 x^{2} + x^{4}

= \int 81 - 18 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 81 d x - \int 18 x^{2} d x + \int x^{4} d x

\int 81 d x = 81 x

\int 18 x^{2} d x = 6 x^{3}

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

= 81 x - 6 x^{3} + \frac{x ^{5}}{5}

定数を解答に追加する

= 81 x - 6 x^{3} + \frac{x ^{5}}{5} + C