integral of (5x^2-x-3)/(x^2(x+1))

解

\int \frac{5 x ^{2} - x - 3}{x ^{2} ( x + 1 )} d x

2 ln ∣ x ∣ + \frac{3}{x} + 3 ln ∣ x + 1 ∣ + C

解答ステップ

\int \frac{5 x ^{2} - x - 3}{x ^{2} ( x + 1 )} d x

以下の部分分数を得る:

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

\int \frac{3}{x + 1} d x = 3 ln ∣ x + 1 ∣

= 2 ln ∣ x ∣ - (- \frac{3}{x}) + 3 ln ∣ x + 1 ∣

簡素化

= 2 ln ∣ x ∣ + \frac{3}{x} + 3 ln ∣ x + 1 ∣

定数を解答に追加する

= 2 ln ∣ x ∣ + \frac{3}{x} + 3 ln ∣ x + 1 ∣ + C