What is the integral of (2x^2-x+1)/(x^2-5x+6) ?

The integral of (2x^2-x+1)/(x^2-5x+6) is

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integral of (2x^2-x+1)/(x^2-5x+6)

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

2 x + \frac{9}{2} ln 4 x^{2} - 20 x + 24 - \frac{21}{2} ln ∣ 2 x - 4 ∣ + \frac{21}{2} ln ∣ 2 x - 6 ∣ - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

Solution steps

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

Expand

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

\int \frac{2 x ^{2}}{x ^{2} - 5 x + 6} d x = 2 (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{x}{x ^{2} - 5 x + 6} d x = 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 24 - \frac{5}{4} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣))

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

= 2 (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) - 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 24 - \frac{5}{4} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣)) - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

Simplify

2 (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) - 2 (\frac{1}{4} ln 4 x^{2} - 20 x + 24 - \frac{5}{4} (ln ∣ 2 x - 4 ∣ - ln ∣ 2 x - 6 ∣)) - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ : 2 x + \frac{9}{2} ln 4 x^{2} - 20 x + 24 - \frac{21}{2} ln ∣ 2 x - 4 ∣ + \frac{21}{2} ln ∣ 2 x - 6 ∣ - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

= 2 x + \frac{9}{2} ln 4 x^{2} - 20 x + 24 - \frac{21}{2} ln ∣ 2 x - 4 ∣ + \frac{21}{2} ln ∣ 2 x - 6 ∣ - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣

Add a constant to the solution

= 2 x + \frac{9}{2} ln 4 x^{2} - 20 x + 24 - \frac{21}{2} ln ∣ 2 x - 4 ∣ + \frac{21}{2} ln ∣ 2 x - 6 ∣ - ln ∣ x - 2 ∣ + ln ∣ x - 3 ∣ + C

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

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

\int \frac{e ^{x}}{cos ^{2} ( e ^{x} )} d x

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

\int \frac{x ^{3} + 1}{( x ^{2} + 1 ) ^{2}} d x

What is the integral of (2x^2-x+1)/(x^2-5x+6) ?
The integral of (2x^2-x+1)/(x^2-5x+6) is