integral of 6/5 x^{5/3}(1+x^{5/6})

解

\int \frac{6}{5} x^{\frac{5}{3}} (1 + x^{\frac{5}{6}}) d x

\frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}}) + C

解答ステップ

\int \frac{6}{5} x^{\frac{5}{3}} (1 + x^{\frac{5}{6}}) d x

定数を除く:

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

= \frac{6}{5} \cdot \int x^{\frac{5}{3}} (1 + x^{\frac{5}{6}}) d x

拡張

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

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

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

\int x^{\frac{5}{3}} d x = \frac{3}{8} x^{\frac{8}{3}}

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

= \frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}})

定数を解答に追加する

= \frac{6}{5} (\frac{3}{8} x^{\frac{8}{3}} + \frac{2}{7} x^{\frac{7}{2}}) + C