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

解

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

- \frac{6}{x} + \frac{5}{x} + C

解答ステップ

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

拡張

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

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

\int \frac{3}{x ^{\frac{3}{2}}} d x = - \frac{6}{x}

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

= - \frac{6}{x} - (- \frac{5}{x})

簡素化

= - \frac{6}{x} + \frac{5}{x}

定数を解答に追加する

= - \frac{6}{x} + \frac{5}{x} + C