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

解

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

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

解答ステップ

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

以下の部分分数を得る:

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

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

\int 1 d x = x

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

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

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

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

定数を解答に追加する

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