integral of sqrt(x)(2/(x^2)-x^3+2x)

解

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

- \frac{4}{x} - \frac{2}{9} x^{\frac{9}{2}} + \frac{4}{5} x^{\frac{5}{2}} + C

解答ステップ

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

拡張

x (\frac{2}{x ^{2}} - x^{3} + 2 x) : \frac{2}{x ^{\frac{3}{2}}} - x^{\frac{7}{2}} + 2 x^{\frac{3}{2}}

= \int \frac{2}{x ^{\frac{3}{2}}} - x^{\frac{7}{2}} + 2 x^{\frac{3}{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

= \int \frac{2}{x ^{\frac{3}{2}}} d x - \int x^{\frac{7}{2}} d x + \int 2 x^{\frac{3}{2}} d x

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

\int x^{\frac{7}{2}} d x = \frac{2}{9} x^{\frac{9}{2}}

\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}}

= - \frac{4}{x} - \frac{2}{9} x^{\frac{9}{2}} + \frac{4}{5} x^{\frac{5}{2}}

定数を解答に追加する

= - \frac{4}{x} - \frac{2}{9} x^{\frac{9}{2}} + \frac{4}{5} x^{\frac{5}{2}} + C