What is the integral of (2x+4)/((x^2-5x+6)) ?

The integral of (2x+4)/((x^2-5x+6)) is -8ln|x-2|+10ln|x-3|+C

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integral of (2x+4)/((x^2-5x+6))

\int \frac{2 x + 4}{( x ^{2} - 5 x + 6 )} d x

- 8 ln ∣ x - 2 ∣ + 10 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{2 x + 4}{x ^{2} - 5 x + 6} d x

Take the partial fraction of

\frac{2 x + 4}{x ^{2} - 5 x + 6} : - \frac{8}{x - 2} + \frac{10}{x - 3}

= \int - \frac{8}{x - 2} + \frac{10}{x - 3} 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{8}{x - 2} d x + \int \frac{10}{x - 3} d x

\int \frac{8}{x - 2} d x = 8 ln ∣ x - 2 ∣

\int \frac{10}{x - 3} d x = 10 ln ∣ x - 3 ∣

= - 8 ln ∣ x - 2 ∣ + 10 ln ∣ x - 3 ∣

Add a constant to the solution

= - 8 ln ∣ x - 2 ∣ + 10 ln ∣ x - 3 ∣ + C

What is the integral of (2x+4)/((x^2-5x+6)) ?
The integral of (2x+4)/((x^2-5x+6)) is -8ln|x-2|+10ln|x-3|+C