integral of (x^2-6)/(x(x-3))

解

\int \frac{x ^{2} - 6}{x ( x - 3 )} d x

x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3 + C

解答ステップ

\int \frac{x ^{2} - 6}{x ( x - 3 )} d x

拡張

\frac{x ^{2} - 6}{x ( x - 3 )} : \frac{x ^{2}}{x ( x - 3 )} - \frac{6}{x ( x - 3 )}

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{x ^{2}}{x ( x - 3 )} d x - \int \frac{6}{x ( x - 3 )} d x

\int \frac{x ^{2}}{x ( x - 3 )} d x = x - 3 + 3 ln ∣ x - 3 ∣

\int \frac{6}{x ( x - 3 )} d x = 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

= x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣)

簡素化

x - 3 + 3 ln ∣ x - 3 ∣ - 6 (- \frac{1}{3} ln ∣ x ∣ + \frac{1}{3} ln ∣ x - 3 ∣) : x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

= x + ln ∣ x - 3 ∣ + 2 ln ∣ x ∣ - 3

定数を解答に追加する

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