What is the integral of (x-2)/(x^2+6x+8) ?

The integral of (x-2)/(x^2+6x+8) is -2ln|x+2|+3ln|x+4|+C

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integral of (x-2)/(x^2+6x+8)

\int \frac{x - 2}{x ^{2} + 6 x + 8} d x

- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 4 ∣ + C

Solution steps

\int \frac{x - 2}{x ^{2} + 6 x + 8} d x

Take the partial fraction of

\frac{x - 2}{x ^{2} + 6 x + 8} : - \frac{2}{x + 2} + \frac{3}{x + 4}

= \int - \frac{2}{x + 2} + \frac{3}{x + 4} d 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{2}{x + 2} d x + \int \frac{3}{x + 4} d x

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

\int \frac{3}{x + 4} d x = 3 ln ∣ x + 4 ∣

= - 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 4 ∣

Add a constant to the solution

= - 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 4 ∣ + C

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What is the integral of (x-2)/(x^2+6x+8) ?
The integral of (x-2)/(x^2+6x+8) is -2ln|x+2|+3ln|x+4|+C