integral of (5-4x^3+2x^6)/(x^6)

解

\int \frac{5 - 4 x ^{3} + 2 x ^{6}}{x ^{6}} d x

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

解答ステップ

\int \frac{5 - 4 x ^{3} + 2 x ^{6}}{x ^{6}} d x

拡張

\frac{5 - 4 x ^{3} + 2 x ^{6}}{x ^{6}} : \frac{5}{x ^{6}} - \frac{4}{x ^{3}} + 2

= \int \frac{5}{x ^{6}} - \frac{4}{x ^{3}} + 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{5}{x ^{6}} d x - \int \frac{4}{x ^{3}} d x + \int 2 d x

\int \frac{5}{x ^{6}} d x = - \frac{1}{x ^{5}}

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

\int 2 d x = 2 x

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

簡素化

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

定数を解答に追加する

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