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

解

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

\frac{2}{3} x^{\frac{3}{2}} + \frac{2}{x ^{\frac{1}{2}}} + C

解答ステップ

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

拡張

\frac{x ^{2} - 1}{x ^{\frac{3}{2}}} : x^{\frac{1}{2}} - \frac{1}{x ^{\frac{3}{2}}}

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

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

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

= \frac{2}{3} x^{\frac{3}{2}} - (- \frac{2}{x ^{\frac{1}{2}}})

簡素化

= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{x ^{\frac{1}{2}}}

定数を解答に追加する

= \frac{2}{3} x^{\frac{3}{2}} + \frac{2}{x ^{\frac{1}{2}}} + C