integral of (a^{2/3}-x^{2/3})^2

解

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

a^{\frac{4}{3}} x - \frac{6}{5} a^{\frac{2}{3}} x^{\frac{5}{3}} + \frac{3}{7} x^{\frac{7}{3}} + C

解答ステップ

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

拡張

(a^{\frac{2}{3}} - x^{\frac{2}{3}})^{2} : a^{\frac{4}{3}} - 2 a^{\frac{2}{3}} x^{\frac{2}{3}} + x^{\frac{4}{3}}

= \int a^{\frac{4}{3}} - 2 a^{\frac{2}{3}} x^{\frac{2}{3}} + x^{\frac{4}{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 a^{\frac{4}{3}} d x - \int 2 a^{\frac{2}{3}} x^{\frac{2}{3}} d x + \int x^{\frac{4}{3}} d x

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

\int 2 a^{\frac{2}{3}} x^{\frac{2}{3}} d x = \frac{6}{5} a^{\frac{2}{3}} x^{\frac{5}{3}}

\int x^{\frac{4}{3}} d x = \frac{3}{7} x^{\frac{7}{3}}

= a^{\frac{4}{3}} x - \frac{6}{5} a^{\frac{2}{3}} x^{\frac{5}{3}} + \frac{3}{7} x^{\frac{7}{3}}

定数を解答に追加する

= a^{\frac{4}{3}} x - \frac{6}{5} a^{\frac{2}{3}} x^{\frac{5}{3}} + \frac{3}{7} x^{\frac{7}{3}} + C