integral of (x+5)/((x+2)(x-1))

解答

\int \frac{x + 5}{( x + 2 ) ( x - 1 )} d x

- ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C

求解步骤

\int \frac{x + 5}{( x + 2 ) ( x - 1 )} d x

将

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

用部份分式展开:

- \frac{1}{x + 2} + \frac{2}{x - 1}

= \int - \frac{1}{x + 2} + \frac{2}{x - 1} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{x + 2} d x + \int \frac{2}{x - 1} d x

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

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

= - ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣

解答补常数

= - ln ∣ x + 2 ∣ + 2 ln ∣ x - 1 ∣ + C