What is the integral of ((x^2-5x+9))/(x^2-5x+6) ?

The integral of ((x^2-5x+9))/(x^2-5x+6) is 3(-ln|x-2|+ln|x-3|)+x+C

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integral of ((x^2-5x+9))/(x^2-5x+6)

\int \frac{( x ^{2} - 5 x + 9 )}{x ^{2} - 5 x + 6} d x

3 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + x + C

Solution steps

\int \frac{x ^{2} - 5 x + 9}{x ^{2} - 5 x + 6} d x

\frac{x ^{2} - 5 x + 9}{x ^{2} - 5 x + 6} = \frac{3}{x ^{2} - 5 x + 6} + 1

= \int \frac{3}{x ^{2} - 5 x + 6} + 1 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{3}{x ^{2} - 5 x + 6} d x + \int 1 d x

\int \frac{3}{x ^{2} - 5 x + 6} d x = 3 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

\int 1 d x = x

= 3 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + x

Add a constant to the solution

= 3 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + x + C

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What is the integral of ((x^2-5x+9))/(x^2-5x+6) ?
The integral of ((x^2-5x+9))/(x^2-5x+6) is 3(-ln|x-2|+ln|x-3|)+x+C