integral of x(7-x)^2

解

\int x (7 - x)^{2} d x

\frac{49 x ^{2}}{2} - \frac{14 x ^{3}}{3} + \frac{x ^{4}}{4} + C

解答ステップ

\int x (7 - x)^{2} d x

拡張

x (7 - x)^{2} : 49 x - 14 x^{2} + x^{3}

= \int 49 x - 14 x^{2} + x^{3} 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 49 x d x - \int 14 x^{2} d x + \int x^{3} d x

\int 49 x d x = \frac{49 x ^{2}}{2}

\int 14 x^{2} d x = \frac{14 x ^{3}}{3}

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

= \frac{49 x ^{2}}{2} - \frac{14 x ^{3}}{3} + \frac{x ^{4}}{4}

定数を解答に追加する

= \frac{49 x ^{2}}{2} - \frac{14 x ^{3}}{3} + \frac{x ^{4}}{4} + C