integral of ((2x+2))/(x^2+10x+24)

解

\int \frac{( 2 x + 2 )}{x ^{2} + 10 x + 24} d x

ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C

解答ステップ

\int \frac{2 x + 2}{x ^{2} + 10 x + 24} d x

拡張

\frac{2 x + 2}{x ^{2} + 10 x + 24} : \frac{2 x}{x ^{2} + 10 x + 24} + \frac{2}{x ^{2} + 10 x + 24}

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{2 x}{x ^{2} + 10 x + 24} d x + \int \frac{2}{x ^{2} + 10 x + 24} d x

\int \frac{2 x}{x ^{2} + 10 x + 24} d x = ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣

\int \frac{2}{x ^{2} + 10 x + 24} d x = ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

= ln x^{2} + 10 x + 24 + 5 ln ∣ x + 6 ∣ - 5 ln ∣ x + 4 ∣ + ln ∣ x + 4 ∣ - ln ∣ x + 6 ∣

簡素化

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣

定数を解答に追加する

= ln x^{2} + 10 x + 24 + 4 ln ∣ x + 6 ∣ - 4 ln ∣ x + 4 ∣ + C