integral of (3x)/((x-1)^2)

解答

\int \frac{3 x}{( x - 1 ) ^{2}} d x

3 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) + C

求解步骤

\int \frac{3 x}{( x - 1 ) ^{2}} d x

提出常数:

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

= 3 \cdot \int \frac{x}{( x - 1 ) ^{2}} d x

使用换元积分法

= 3 \cdot \int \frac{u + 1}{u ^{2}} d u

乘开

\frac{u + 1}{u ^{2}} : \frac{1}{u} + \frac{1}{u ^{2}}

= 3 \cdot \int \frac{1}{u} + \frac{1}{u ^{2}} 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

= 3 (\int \frac{1}{u} d u + \int \frac{1}{u ^{2}} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u ^{2}} d u = - \frac{1}{u}

= 3 (ln ∣ u ∣ - \frac{1}{u})

u = x - 1

代回

= 3 (ln ∣ x - 1 ∣ - \frac{1}{x - 1})

解答补常数

= 3 (ln ∣ x - 1 ∣ - \frac{1}{x - 1}) + C

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