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

解答

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

2 ln ∣ x + 1 ∣ + C

求解步骤

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

乘开

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

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

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

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

化简

2 (ln ∣ x + 1 ∣ + \frac{1}{x + 1}) - \frac{2}{x + 1} : 2 ln ∣ x + 1 ∣

= 2 ln ∣ x + 1 ∣

解答补常数

= 2 ln ∣ x + 1 ∣ + C

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