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

解

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

- \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}} + C

解答ステップ

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

拡張

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

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

\int \frac{x ^{3}}{( x - 3 ) ^{3}} d x = - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )}

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

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

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} - (- \frac{1}{2 ( x - 3 ) ^{2}})

簡素化

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}}

定数を解答に追加する

= - \frac{x ^{3}}{2 ( x - 3 ) ^{2}} + \frac{3 ( x ^{2} - 12 x + 18 + 6 x ln ∣ x - 3 ∣ - 18 ln ∣ x - 3 ∣ )}{2 ( x - 3 )} - \frac{2}{x - 3} - \frac{3}{( x - 3 ) ^{2}} + \frac{1}{2 ( x - 3 ) ^{2}} + C