integral of (x+5)/(x^2+10x)

解

\int \frac{x + 5}{x ^{2} + 10 x} d x

\frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣ + C

解答ステップ

\int \frac{x + 5}{x ^{2} + 10 x} d x

以下の部分分数を得る:

\frac{x + 5}{x ^{2} + 10 x} : \frac{1}{2 x} + \frac{1}{2 ( x + 10 )}

= \int \frac{1}{2 x} + \frac{1}{2 ( x + 10 )} 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{1}{2 x} d x + \int \frac{1}{2 ( x + 10 )} d x

\int \frac{1}{2 x} d x = \frac{1}{2} ln ∣ x ∣

\int \frac{1}{2 ( x + 10 )} d x = \frac{1}{2} ln ∣ x + 10 ∣

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣

定数を解答に追加する

= \frac{1}{2} ln ∣ x ∣ + \frac{1}{2} ln ∣ x + 10 ∣ + C