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

解

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

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

解答ステップ

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

拡張

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

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

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

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

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

簡素化

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

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

定数を解答に追加する

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