積分記号 (x^3+4x^2-3x+2)/(x^2)

解

積分記号

\frac{x ^{3} + 4 x ^{2} - 3 x + 2}{x ^{2}}

\frac{x ^{2}}{2} + 4 x - 3 ln ∣ x ∣ - \frac{2}{x} + C

解答ステップ

\int \frac{x ^{3} + 4 x ^{2} - 3 x + 2}{x ^{2}} d x

拡張

\frac{x ^{3} + 4 x ^{2} - 3 x + 2}{x ^{2}} : x + 4 - \frac{3}{x} + \frac{2}{x ^{2}}

= \int x + 4 - \frac{3}{x} + \frac{2}{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 x d x + \int 4 d x - \int \frac{3}{x} d x + \int \frac{2}{x ^{2}} d x

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

\int 4 d x = 4 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

= \frac{x ^{2}}{2} + 4 x - 3 ln ∣ x ∣ - \frac{2}{x}

定数を解答に追加する

= \frac{x ^{2}}{2} + 4 x - 3 ln ∣ x ∣ - \frac{2}{x} + C