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

解

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

4 ln ∣ x + 2 ∣ + ln ∣ x ∣ + \frac{2}{x} + C

解答ステップ

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

拡張

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

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

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

\int \frac{4}{x ^{2} ( x + 2 )} d x = - ln ∣ x ∣ - \frac{2}{x} + ln ∣ x + 2 ∣

= 5 ln ∣ x + 2 ∣ - (- ln ∣ x ∣ - \frac{2}{x} + ln ∣ x + 2 ∣)

簡素化

= 4 ln ∣ x + 2 ∣ + ln ∣ x ∣ + \frac{2}{x}

定数を解答に追加する

= 4 ln ∣ x + 2 ∣ + ln ∣ x ∣ + \frac{2}{x} + C