integral of (x^2+12x+12)/(x(x^2-4))

解

\int \frac{x ^{2} + 12 x + 12}{x ( x ^{2} - 4 )} d x

- 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C

解答ステップ

\int \frac{x ^{2} + 12 x + 12}{x ( x ^{2} - 4 )} d x

以下の部分分数を得る:

\frac{x ^{2} + 12 x + 12}{x ( x ^{2} - 4 )} : - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{x - 2}

= \int - \frac{3}{x} - \frac{1}{x + 2} + \frac{5}{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 \frac{3}{x} d x - \int \frac{1}{x + 2} d x + \int \frac{5}{x - 2} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int \frac{1}{x + 2} d x = ln ∣ x + 2 ∣

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

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣

定数を解答に追加する

= - 3 ln ∣ x ∣ - ln ∣ x + 2 ∣ + 5 ln ∣ x - 2 ∣ + C