integral of (x^3-5x^2-sqrt(x))/(x^2)

解

\int \frac{x ^{3} - 5 x ^{2} - x}{x ^{2}} d x

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

解答ステップ

\int \frac{x ^{3} - 5 x ^{2} - x}{x ^{2}} d x

拡張

\frac{x ^{3} - 5 x ^{2} - x}{x ^{2}} : x - 5 - \frac{1}{x ^{\frac{3}{2}}}

= \int x - 5 - \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 d x - \int 5 d x - \int \frac{1}{x ^{\frac{3}{2}}} d x

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

\int 5 d x = 5 x

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

= \frac{x ^{2}}{2} - 5 x - (- \frac{2}{x})

簡素化

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

定数を解答に追加する

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