What is the integral of (x^2-9x+25)/(x^2-9x+20) ?

The integral of (x^2-9x+25)/(x^2-9x+20) is 5(-ln|x-4|+ln|x-5|)+x+C

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integral of (x^2-9x+25)/(x^2-9x+20)

\int \frac{x ^{2} - 9 x + 25}{x ^{2} - 9 x + 20} d x

5 (- ln ∣ x - 4 ∣ + ln ∣ x - 5 ∣) + x + C

Solution steps

\int \frac{x ^{2} - 9 x + 25}{x ^{2} - 9 x + 20} d x

\frac{x ^{2} - 9 x + 25}{x ^{2} - 9 x + 20} = \frac{5}{x ^{2} - 9 x + 20} + 1

= \int \frac{5}{x ^{2} - 9 x + 20} + 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{5}{x ^{2} - 9 x + 20} d x + \int 1 d x

\int \frac{5}{x ^{2} - 9 x + 20} d x = 5 (- ln ∣ x - 4 ∣ + ln ∣ x - 5 ∣)

\int 1 d x = x

= 5 (- ln ∣ x - 4 ∣ + ln ∣ x - 5 ∣) + x

Add a constant to the solution

= 5 (- ln ∣ x - 4 ∣ + ln ∣ x - 5 ∣) + x + C

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What is the integral of (x^2-9x+25)/(x^2-9x+20) ?
The integral of (x^2-9x+25)/(x^2-9x+20) is 5(-ln|x-4|+ln|x-5|)+x+C