integral of x(5-x^2)

解

\int x (5 - x^{2}) d x

\frac{5 x ^{2}}{2} - \frac{x ^{4}}{4} + C

解答ステップ

\int x (5 - x^{2}) d x

拡張

x (5 - x^{2}) : 5 x - x^{3}

= \int 5 x - 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 5 x d x - \int x^{3} d x

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

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

= \frac{5 x ^{2}}{2} - \frac{x ^{4}}{4}

定数を解答に追加する

= \frac{5 x ^{2}}{2} - \frac{x ^{4}}{4} + C