integral of (x^4-6)/(x^3-2x^2)

解

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

\frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C

解答ステップ

\int \frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}} d x

因数

\frac{x ^{4} - 6}{x ^{3} - 2 x ^{2}}

= \int \frac{( x + 4 6 ) ( x - 4 6 ) ( x ^{2} + 6 )}{x ^{2} ( x - 2 )} d x

以下の部分分数を得る:

\frac{( x + 4 6 ) ( x - 4 6 ) ( x ^{2} + 6 )}{x ^{2} ( x - 2 )} : x + 2 + \frac{3}{2 x} + \frac{3}{x ^{2}} + \frac{5}{2 ( x - 2 )}

= \int x + 2 + \frac{3}{2 x} + \frac{3}{x ^{2}} + \frac{5}{2 ( x - 2 )} 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 x d x + \int 2 d x + \int \frac{3}{2 x} d x + \int \frac{3}{x ^{2}} d x + \int \frac{5}{2 ( x - 2 )} d x

\int x d x = \frac{x ^{2}}{2}

\int 2 d x = 2 x

\int \frac{3}{2 x} d x = \frac{3}{2} ln ∣ x ∣

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

\int \frac{5}{2 ( x - 2 )} d x = \frac{5}{2} ln ∣ x - 2 ∣

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣

定数を解答に追加する

= \frac{x ^{2}}{2} + 2 x + \frac{3}{2} ln ∣ x ∣ - \frac{3}{x} + \frac{5}{2} ln ∣ x - 2 ∣ + C