integral of x/(7x+2)

解答

\int \frac{x}{7 x + 2} d x

\frac{1}{49} (7 x + 2 - 2 ln ∣ 7 x + 2 ∣) + C

求解步骤

\int \frac{x}{7 x + 2} d x

使用换元积分法

= \int \frac{u - 2}{49 u} d u

提出常数:

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

= \frac{1}{49} \cdot \int \frac{u - 2}{u} d u

乘开

\frac{u - 2}{u} : 1 - \frac{2}{u}

= \frac{1}{49} \cdot \int 1 - \frac{2}{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

= \frac{1}{49} (\int 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= \frac{1}{49} (u - 2 ln ∣ u ∣)

u = 7 x + 2

代回

= \frac{1}{49} (7 x + 2 - 2 ln ∣ 7 x + 2 ∣)

解答补常数

= \frac{1}{49} (7 x + 2 - 2 ln ∣ 7 x + 2 ∣) + C

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