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

解

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

\frac{3}{13} x^{\frac{13}{3}} - \frac{3}{7} x^{\frac{7}{3}} - 6 x^{\frac{1}{3}} + C

解答ステップ

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

拡張

\frac{( x ^{2} + 1 ) ( x ^{2} - 2 )}{x ^{\frac{2}{3}}} : x^{\frac{10}{3}} - x^{\frac{4}{3}} - \frac{2}{x ^{\frac{2}{3}}}

= \int x^{\frac{10}{3}} - x^{\frac{4}{3}} - \frac{2}{x ^{\frac{2}{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 x^{\frac{10}{3}} d x - \int x^{\frac{4}{3}} d x - \int \frac{2}{x ^{\frac{2}{3}}} d x

\int x^{\frac{10}{3}} d x = \frac{3}{13} x^{\frac{13}{3}}

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

\int \frac{2}{x ^{\frac{2}{3}}} d x = 6 x^{\frac{1}{3}}

= \frac{3}{13} x^{\frac{13}{3}} - \frac{3}{7} x^{\frac{7}{3}} - 6 x^{\frac{1}{3}}

定数を解答に追加する

= \frac{3}{13} x^{\frac{13}{3}} - \frac{3}{7} x^{\frac{7}{3}} - 6 x^{\frac{1}{3}} + C