integral of (x-3)/(x^2-7x-18)

解

\int \frac{x - 3}{x ^{2} - 7 x - 18} d x

\frac{5}{11} ln ∣ x + 2 ∣ + \frac{6}{11} ln ∣ x - 9 ∣ + C

解答ステップ

\int \frac{x - 3}{x ^{2} - 7 x - 18} d x

以下の部分分数を得る:

\frac{x - 3}{x ^{2} - 7 x - 18} : \frac{5}{11 ( x + 2 )} + \frac{6}{11 ( x - 9 )}

= \int \frac{5}{11 ( x + 2 )} + \frac{6}{11 ( x - 9 )} 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}{11 ( x + 2 )} d x + \int \frac{6}{11 ( x - 9 )} d x

\int \frac{5}{11 ( x + 2 )} d x = \frac{5}{11} ln ∣ x + 2 ∣

\int \frac{6}{11 ( x - 9 )} d x = \frac{6}{11} ln ∣ x - 9 ∣

= \frac{5}{11} ln ∣ x + 2 ∣ + \frac{6}{11} ln ∣ x - 9 ∣

定数を解答に追加する

= \frac{5}{11} ln ∣ x + 2 ∣ + \frac{6}{11} ln ∣ x - 9 ∣ + C