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

解

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

- \frac{2}{x} - ln ∣ x ∣ + C

解答ステップ

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

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

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

簡素化

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

= - \frac{2}{x} - ln ∣ x ∣

定数を解答に追加する

= - \frac{2}{x} - ln ∣ x ∣ + C