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

解答

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

ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

求解步骤

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

乘开

\frac{x + 2}{x ^{2} - 2 x + 1} : \frac{x}{x ^{2} - 2 x + 1} + \frac{2}{x ^{2} - 2 x + 1}

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

\int \frac{x}{x ^{2} - 2 x + 1} d x = ln ∣ x - 1 ∣ - \frac{1}{x - 1}

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

= ln ∣ x - 1 ∣ - \frac{1}{x - 1} - \frac{2}{x - 1}

化简

ln ∣ x - 1 ∣ - \frac{1}{x - 1} - \frac{2}{x - 1} : ln ∣ x - 1 ∣ - \frac{3}{x - 1}

= ln ∣ x - 1 ∣ - \frac{3}{x - 1}

解答补常数

= ln ∣ x - 1 ∣ - \frac{3}{x - 1} + C

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