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

解

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

x - 3 ln ∣ x ∣ - \frac{2}{x} + C

解答ステップ

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

拡張

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

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= x - 3 ln ∣ x ∣ - \frac{2}{x}

定数を解答に追加する

= x - 3 ln ∣ x ∣ - \frac{2}{x} + C