integral of (1-1/x)^2 1/(x^2)

解

\int (1 - \frac{1}{x})^{2} \frac{1}{x ^{2}} d x

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

解答ステップ

\int (1 - \frac{1}{x})^{2} \frac{1}{x ^{2}} d x

部分積分を適用する

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

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

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

簡素化

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

定数を解答に追加する

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