integral of (x-x^3)/(sqrt(x))

解

\int \frac{x - x ^{3}}{x} d x

\frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C

解答ステップ

\int \frac{x - x ^{3}}{x} d x

簡素化

\frac{x - x ^{3}}{x} : x (1 - x^{2})

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

拡張

x (1 - x^{2}) : x - x^{\frac{5}{2}}

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

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

\int x^{\frac{5}{2}} d x = \frac{2}{7} x^{\frac{7}{2}}

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}}

定数を解答に追加する

= \frac{2}{3} x^{\frac{3}{2}} - \frac{2}{7} x^{\frac{7}{2}} + C