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

解

\int \frac{x + 2}{x ^{2} + 5 x} d x

\frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C

解答ステップ

\int \frac{x + 2}{x ^{2} + 5 x} d x

拡張

\frac{x + 2}{x ^{2} + 5 x} : \frac{x}{x ^{2} + 5 x} + \frac{2}{x ^{2} + 5 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{x}{x ^{2} + 5 x} d x + \int \frac{2}{x ^{2} + 5 x} d x

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

\int \frac{2}{x ^{2} + 5 x} d x = 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

= ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣)

簡素化

ln ∣ x + 5 ∣ + 2 (\frac{1}{5} ln ∣ x ∣ - \frac{1}{5} ln ∣ x + 5 ∣) : \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣

定数を解答に追加する

= \frac{2}{5} ln ∣ x ∣ + \frac{3}{5} ln ∣ x + 5 ∣ + C