integral of (x-4)/(x^9)

解

\int \frac{x - 4}{x ^{9}} d x

- \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}} + C

解答ステップ

\int \frac{x - 4}{x ^{9}} d x

拡張

\frac{x - 4}{x ^{9}} : \frac{1}{x ^{8}} - \frac{4}{x ^{9}}

= \int \frac{1}{x ^{8}} - \frac{4}{x ^{9}} 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{1}{x ^{8}} d x - \int \frac{4}{x ^{9}} d x

\int \frac{1}{x ^{8}} d x = - \frac{1}{7 x ^{7}}

\int \frac{4}{x ^{9}} d x = - \frac{1}{2 x ^{8}}

= - \frac{1}{7 x ^{7}} - (- \frac{1}{2 x ^{8}})

簡素化

= - \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}}

定数を解答に追加する

= - \frac{1}{7 x ^{7}} + \frac{1}{2 x ^{8}} + C