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

解答

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

3 ln ∣ x - 1 ∣ - \frac{1}{x - 1} + C

求解步骤

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

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

\int \frac{3}{x - 1} d x = 3 ln ∣ x - 1 ∣

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

= 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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