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

解

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

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

解答ステップ

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

拡張

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

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

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

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

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

定数を解答に追加する

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