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

The integral of (2x+5)/(x^2-5x+6) is ln|4x^2-20x+24|-5ln|2x-4|+5ln|2x-6|+5(-ln|x-2|+ln|x-3|)+C

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

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

ln 4 x^{2} - 20 x + 24 - 5 ln ∣ 2 x - 4 ∣ + 5 ln ∣ 2 x - 6 ∣ + 5 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

Solution steps

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

Expand

\frac{2 x + 5}{x ^{2} - 5 x + 6} : \frac{2 x}{x ^{2} - 5 x + 6} + \frac{5}{x ^{2} - 5 x + 6}

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

\int \frac{2 x}{x ^{2} - 5 x + 6} d x = ln 4 x^{2} - 20 x + 24 - 5 ln ∣ 2 x - 4 ∣ + 5 ln ∣ 2 x - 6 ∣

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

= ln 4 x^{2} - 20 x + 24 - 5 ln ∣ 2 x - 4 ∣ + 5 ln ∣ 2 x - 6 ∣ + 5 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

Add a constant to the solution

= ln 4 x^{2} - 20 x + 24 - 5 ln ∣ 2 x - 4 ∣ + 5 ln ∣ 2 x - 6 ∣ + 5 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

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What is the integral of (2x+5)/(x^2-5x+6) ?
The integral of (2x+5)/(x^2-5x+6) is ln|4x^2-20x+24|-5ln|2x-4|+5ln|2x-6|+5(-ln|x-2|+ln|x-3|)+C