integral of (2x+3)/(x^2-5x+4)

解

\int \frac{2 x + 3}{x ^{2} - 5 x + 4} d x

- \frac{5}{3} ln ∣ x - 1 ∣ + \frac{11}{3} ln ∣ x - 4 ∣ + C

解答ステップ

\int \frac{2 x + 3}{x ^{2} - 5 x + 4} d x

以下の部分分数を得る:

\frac{2 x + 3}{x ^{2} - 5 x + 4} : - \frac{5}{3 ( x - 1 )} + \frac{11}{3 ( x - 4 )}

= \int - \frac{5}{3 ( x - 1 )} + \frac{11}{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{5}{3 ( x - 1 )} d x + \int \frac{11}{3 ( x - 4 )} d x

\int \frac{5}{3 ( x - 1 )} d x = \frac{5}{3} ln ∣ x - 1 ∣

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

= - \frac{5}{3} ln ∣ x - 1 ∣ + \frac{11}{3} ln ∣ x - 4 ∣

定数を解答に追加する

= - \frac{5}{3} ln ∣ x - 1 ∣ + \frac{11}{3} ln ∣ x - 4 ∣ + C