integral of (2x^4+2x^3-x)/(x^3)

解

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

x^{2} + 2 x + \frac{1}{x} + C

解答ステップ

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

拡張

\frac{2 x ^{4} + 2 x ^{3} - x}{x ^{3}} : 2 x + 2 - \frac{1}{x ^{2}}

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

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

\int 2 d x = 2 x

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

= x^{2} + 2 x - (- \frac{1}{x})

簡素化

= x^{2} + 2 x + \frac{1}{x}

定数を解答に追加する

= x^{2} + 2 x + \frac{1}{x} + C