integral of (x^2-3)/(x^3-3x^2+3x-1)

解

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

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

解答ステップ

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

拡張

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

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

\int \frac{x ^{2}}{x ^{3} - 3 x ^{2} + 3 x - 1} d x = \frac{x ^{3}}{3 ( x ^{3} - 3 x ^{2} + 3 x - 1 )} + ln ∣ x - 1 ∣ - \frac{3}{x - 1} - \frac{3}{2 ( x - 1 ) ^{2}} - \frac{1}{3 ( x - 1 ) ^{3}}

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

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

簡素化

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

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

定数を解答に追加する

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