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

解

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

x^{2} + x - ln ∣ x ∣ + C

解答ステップ

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

拡張

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

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

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

\int 1 d x = x

\int \frac{1}{x} d x = ln ∣ x ∣

= x^{2} + x - ln ∣ x ∣

定数を解答に追加する

= x^{2} + x - ln ∣ x ∣ + C