integral of (1-x^3)^2x

解

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

\frac{x ^{2}}{2} - \frac{2 x ^{5}}{5} + \frac{x ^{8}}{8} + C

解答ステップ

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

拡張

(1 - x^{3})^{2} x : x - 2 x^{4} + x^{7}

= \int x - 2 x^{4} + x^{7} 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 d x - \int 2 x^{4} d x + \int x^{7} d x

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

\int 2 x^{4} d x = \frac{2 x ^{5}}{5}

\int x^{7} d x = \frac{x ^{8}}{8}

= \frac{x ^{2}}{2} - \frac{2 x ^{5}}{5} + \frac{x ^{8}}{8}

定数を解答に追加する

= \frac{x ^{2}}{2} - \frac{2 x ^{5}}{5} + \frac{x ^{8}}{8} + C