What is the integral of (5x-23)/(x^2-9x+20) ?

The integral of (5x-23)/(x^2-9x+20) is 3ln|x-4|+2ln|x-5|+C

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integral of (5x-23)/(x^2-9x+20)

\int \frac{5 x - 23}{x ^{2} - 9 x + 20} d x

3 ln ∣ x - 4 ∣ + 2 ln ∣ x - 5 ∣ + C

Solution steps

\int \frac{5 x - 23}{x ^{2} - 9 x + 20} d x

Take the partial fraction of

\frac{5 x - 23}{x ^{2} - 9 x + 20} : \frac{3}{x - 4} + \frac{2}{x - 5}

= \int \frac{3}{x - 4} + \frac{2}{x - 5} 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 - 4} d x + \int \frac{2}{x - 5} d x

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

\int \frac{2}{x - 5} d x = 2 ln ∣ x - 5 ∣

= 3 ln ∣ x - 4 ∣ + 2 ln ∣ x - 5 ∣

Add a constant to the solution

= 3 ln ∣ x - 4 ∣ + 2 ln ∣ x - 5 ∣ + C

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What is the integral of (5x-23)/(x^2-9x+20) ?
The integral of (5x-23)/(x^2-9x+20) is 3ln|x-4|+2ln|x-5|+C