integral of (x-44)/(x^2(x+4))

解

\int \frac{x - 44}{x ^{2} ( x + 4 )} d x

3 ln ∣ x ∣ + \frac{11}{x} - 3 ln ∣ x + 4 ∣ + C

解答ステップ

\int \frac{x - 44}{x ^{2} ( x + 4 )} d x

以下の部分分数を得る:

\frac{x - 44}{x ^{2} ( x + 4 )} : \frac{3}{x} - \frac{11}{x ^{2}} - \frac{3}{x + 4}

= \int \frac{3}{x} - \frac{11}{x ^{2}} - \frac{3}{x + 4} 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{11}{x ^{2}} d x - \int \frac{3}{x + 4} d x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

\int \frac{11}{x ^{2}} d x = - \frac{11}{x}

\int \frac{3}{x + 4} d x = 3 ln ∣ x + 4 ∣

= 3 ln ∣ x ∣ - (- \frac{11}{x}) - 3 ln ∣ x + 4 ∣

簡素化

= 3 ln ∣ x ∣ + \frac{11}{x} - 3 ln ∣ x + 4 ∣

定数を解答に追加する

= 3 ln ∣ x ∣ + \frac{11}{x} - 3 ln ∣ x + 4 ∣ + C