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受欢迎的微积分 >

integral of (4x)/(x-8)

解答

\int \frac{4 x}{x - 8} d x

解答

4 (x - 8 + 8 ln ∣ x - 8 ∣) + C

求解步骤

\int \frac{4 x}{x - 8} d x

提出常数:

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

= 4 \cdot \int \frac{x}{x - 8} d x

使用换元积分法

= 4 \cdot \int \frac{u + 8}{u} d u

乘开

\frac{u + 8}{u} : 1 + \frac{8}{u}

= 4 \cdot \int 1 + \frac{8}{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

= 4 (\int 1 d u + \int \frac{8}{u} d u)

\int 1 d u = u

\int \frac{8}{u} d u = 8 ln ∣ u ∣

= 4 (u + 8 ln ∣ u ∣)

u = x - 8

代回

= 4 (x - 8 + 8 ln ∣ x - 8 ∣)

解答补常数

= 4 (x - 8 + 8 ln ∣ x - 8 ∣) + C

作图

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