What is the integral of (-x+2)/(x^2+2x) ?

The integral of (-x+2)/(x^2+2x) is -2ln|x+2|+ln|x|+C

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

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

- 2 ln ∣ x + 2 ∣ + ln ∣ x ∣ + C

Solution steps

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

Expand

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

Apply the Sum Rule:

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

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

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

= - ln ∣ x + 2 ∣ - ln ∣ x + 2 ∣ + ln ∣ x ∣

Simplify

= - 2 ln ∣ x + 2 ∣ + ln ∣ x ∣

Add a constant to the solution

= - 2 ln ∣ x + 2 ∣ + ln ∣ x ∣ + C

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What is the integral of (-x+2)/(x^2+2x) ?
The integral of (-x+2)/(x^2+2x) is -2ln|x+2|+ln|x|+C