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

解

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

x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}} + C

解答ステップ

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

拡張

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

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

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

= x - 5 ln ∣ x ∣ - (- \frac{1}{x ^{2}})

簡素化

= x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}}

定数を解答に追加する

= x - 5 ln ∣ x ∣ + \frac{1}{x ^{2}} + C