What is the integral of (x^2+8)/(x^2-5x+6) ?

The integral of (x^2+8)/(x^2-5x+6) is

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integral of (x^2+8)/(x^2-5x+6)

\int \frac{x ^{2} + 8}{x ^{2} - 5 x + 6} d x

4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

Solution steps

\int \frac{x ^{2} + 8}{x ^{2} - 5 x + 6} d x

Expand

\frac{x ^{2} + 8}{x ^{2} - 5 x + 6} : \frac{x ^{2}}{x ^{2} - 5 x + 6} + \frac{8}{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{x ^{2}}{x ^{2} - 5 x + 6} d x + \int \frac{8}{x ^{2} - 5 x + 6} d x

\int \frac{x ^{2}}{x ^{2} - 5 x + 6} d x = 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x

\int \frac{8}{x ^{2} - 5 x + 6} d x = 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣)

Add a constant to the solution

= 4 (\frac{5}{8} ln 4 x^{2} - 20 x + 24 - \frac{13}{8} ln ∣ 2 x - 4 ∣ + \frac{13}{8} ln ∣ 2 x - 6 ∣) + x + 8 (- ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣) + C

\int e^{- a x} cos (w x) d x

\int \frac{1}{2 + cos ( 2 x )} d x

\int \frac{1}{x ln ( a )} d x

\int \frac{2 x - 1}{x - x ^{2}} d x

\int \frac{2 x}{x + 3} d x

What is the integral of (x^2+8)/(x^2-5x+6) ?
The integral of (x^2+8)/(x^2-5x+6) is