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

解答

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

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

求解步骤

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

乘开

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

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

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

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

化简

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

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

解答补常数

= - \frac{x ^{2}}{x + 1} + 2 x + 2 + \frac{2}{x + 1} + C

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