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

解

\int x (x - 1) (x - 2) d x

\frac{x ^{4}}{4} - x^{3} + x^{2} + C

解答ステップ

\int x (x - 1) (x - 2) d x

拡張

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

= \int x^{3} - 3 x^{2} + 2 x 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^{3} d x - \int 3 x^{2} d x + \int 2 x d x

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

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

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

= \frac{x ^{4}}{4} - x^{3} + x^{2}

定数を解答に追加する

= \frac{x ^{4}}{4} - x^{3} + x^{2} + C