integral of 1/(12+4x-x^2)

解

\int \frac{1}{12 + 4 x - x ^{2}} d x

\frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣ + C

解答ステップ

\int \frac{1}{12 + 4 x - x ^{2}} d x

以下の部分分数を得る:

\frac{1}{12 + 4 x - x ^{2}} : \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 6 )}

= \int \frac{1}{8 ( x + 2 )} - \frac{1}{8 ( x - 6 )} 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}{8 ( x + 2 )} d x - \int \frac{1}{8 ( x - 6 )} d x

\int \frac{1}{8 ( x + 2 )} d x = \frac{1}{8} ln ∣ x + 2 ∣

\int \frac{1}{8 ( x - 6 )} d x = \frac{1}{8} ln ∣ x - 6 ∣

= \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣

定数を解答に追加する

= \frac{1}{8} ln ∣ x + 2 ∣ - \frac{1}{8} ln ∣ x - 6 ∣ + C