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

解

\int \frac{( x - 2 ) ^{2}}{x ^{4}} d x

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

解答ステップ

\int \frac{( x - 2 ) ^{2}}{x ^{4}} d x

拡張

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

= \int \frac{1}{x ^{2}} - \frac{4}{x ^{3}} + \frac{4}{x ^{4}} 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 ^{2}} d x - \int \frac{4}{x ^{3}} d x + \int \frac{4}{x ^{4}} d x

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

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

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

= - \frac{1}{x} - (- \frac{2}{x ^{2}}) - \frac{4}{3 x ^{3}}

簡素化

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

定数を解答に追加する

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