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

解

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

2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}}) + C

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (x + 2 x^{\frac{1}{3}}) d x

拡張

x (x + 2 x^{\frac{1}{3}}) : x^{\frac{3}{2}} + 2 x^{\frac{5}{6}}

= 2 \cdot \int x^{\frac{3}{2}} + 2 x^{\frac{5}{6}} 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

= 2 (\int x^{\frac{3}{2}} d x + \int 2 x^{\frac{5}{6}} d x)

\int x^{\frac{3}{2}} d x = \frac{2}{5} x^{\frac{5}{2}}

\int 2 x^{\frac{5}{6}} d x = \frac{12}{11} x^{\frac{11}{6}}

= 2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}})

定数を解答に追加する

= 2 (\frac{2}{5} x^{\frac{5}{2}} + \frac{12}{11} x^{\frac{11}{6}}) + C