integral of (x+1)/(x^5)

解

\int \frac{x + 1}{x ^{5}} d x

- \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}} + C

解答ステップ

\int \frac{x + 1}{x ^{5}} d x

拡張

\frac{x + 1}{x ^{5}} : \frac{1}{x ^{4}} + \frac{1}{x ^{5}}

= \int \frac{1}{x ^{4}} + \frac{1}{x ^{5}} 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 ^{4}} d x + \int \frac{1}{x ^{5}} d x

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{1}{x ^{5}} d x = - \frac{1}{4 x ^{4}}

= - \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}}

定数を解答に追加する

= - \frac{1}{3 x ^{3}} - \frac{1}{4 x ^{4}} + C