integral of x/((1-x)^4)

解

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

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

解答ステップ

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

以下の部分分数を得る:

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

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

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

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

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

定数を解答に追加する

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