integral of x/(x-5)

解答

\int \frac{x}{x - 5} d x

x - 5 + 5 ln ∣ x - 5 ∣ + C

求解步骤

\int \frac{x}{x - 5} d x

使用换元积分法

= \int \frac{u + 5}{u} d u

乘开

\frac{u + 5}{u} : 1 + \frac{5}{u}

= \int 1 + \frac{5}{u} d u

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 1 d u + \int \frac{5}{u} d u

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= u + 5 ln ∣ u ∣

u = x - 5

代回

= x - 5 + 5 ln ∣ x - 5 ∣

解答补常数

= x - 5 + 5 ln ∣ x - 5 ∣ + C