integral of (9x+6)/((x-6)(x+4))

解答

\int \frac{9 x + 6}{( x - 6 ) ( x + 4 )} d x

6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C

求解步骤

\int \frac{9 x + 6}{( x - 6 ) ( x + 4 )} d x

乘开

\frac{9 x + 6}{( x - 6 ) ( x + 4 )} : \frac{9 x}{( x - 6 ) ( x + 4 )} + \frac{6}{( x - 6 ) ( x + 4 )}

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{9 x}{( x - 6 ) ( x + 4 )} d x + \int \frac{6}{( x - 6 ) ( x + 4 )} d x

\int \frac{9 x}{( x - 6 ) ( x + 4 )} d x = 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣)

\int \frac{6}{( x - 6 ) ( x + 4 )} d x = 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

= 9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣)

化简

9 (\frac{3}{5} ln ∣ x - 6 ∣ + \frac{2}{5} ln ∣ x + 4 ∣) + 6 (\frac{1}{10} ln ∣ x - 6 ∣ - \frac{1}{10} ln ∣ x + 4 ∣) : 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣

解答补常数

= 6 ln ∣ x - 6 ∣ + 3 ln ∣ x + 4 ∣ + C