integral of (2x)/(x-4)

解答

\int \frac{2 x}{x - 4} d x

2 (x - 4 + 4 ln ∣ x - 4 ∣) + C

求解步骤

\int \frac{2 x}{x - 4} d x

提出常数:

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

= 2 \cdot \int \frac{x}{x - 4} d x

使用换元积分法

= 2 \cdot \int \frac{u + 4}{u} d u

乘开

\frac{u + 4}{u} : 1 + \frac{4}{u}

= 2 \cdot \int 1 + \frac{4}{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

= 2 (\int 1 d u + \int \frac{4}{u} d u)

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= 2 (u + 4 ln ∣ u ∣)

u = x - 4

代回

= 2 (x - 4 + 4 ln ∣ x - 4 ∣)

解答补常数

= 2 (x - 4 + 4 ln ∣ x - 4 ∣) + C

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