integral of (2x)/(x^2-3x-10)

解

\int \frac{2 x}{x ^{2} - 3 x - 10} d x

2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣) + C

解答ステップ

\int \frac{2 x}{x ^{2} - 3 x - 10} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{x}{x ^{2} - 3 x - 10} d x

以下の部分分数を得る:

\frac{x}{x ^{2} - 3 x - 10} : \frac{2}{7 ( x + 2 )} + \frac{5}{7 ( x - 5 )}

= 2 \cdot \int \frac{2}{7 ( x + 2 )} + \frac{5}{7 ( x - 5 )} 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

= 2 (\int \frac{2}{7 ( x + 2 )} d x + \int \frac{5}{7 ( x - 5 )} d x)

\int \frac{2}{7 ( x + 2 )} d x = \frac{2}{7} ln ∣ x + 2 ∣

\int \frac{5}{7 ( x - 5 )} d x = \frac{5}{7} ln ∣ x - 5 ∣

= 2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣)

定数を解答に追加する

= 2 (\frac{2}{7} ln ∣ x + 2 ∣ + \frac{5}{7} ln ∣ x - 5 ∣) + C