integral of sqrt(x)(x^{1/3}-x^{2/5})

解

\int x (x^{\frac{1}{3}} - x^{\frac{2}{5}}) d x

\frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}} + C

解答ステップ

\int x (x^{\frac{1}{3}} - x^{\frac{2}{5}}) d x

拡張

x (x^{\frac{1}{3}} - x^{\frac{2}{5}}) : x^{\frac{5}{6}} - x^{\frac{9}{10}}

= \int x^{\frac{5}{6}} - x^{\frac{9}{10}} 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^{\frac{5}{6}} d x - \int x^{\frac{9}{10}} d x

\int x^{\frac{5}{6}} d x = \frac{6}{11} x^{\frac{11}{6}}

\int x^{\frac{9}{10}} d x = \frac{10}{19} x^{\frac{19}{10}}

= \frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}}

定数を解答に追加する

= \frac{6}{11} x^{\frac{11}{6}} - \frac{10}{19} x^{\frac{19}{10}} + C