What is the integral of (x^2+3)/(x^2-3x+2) ?

The integral of (x^2+3)/(x^2-3x+2) is x-4ln|x-1|+7ln|x-2|+C

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integral of (x^2+3)/(x^2-3x+2)

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

x - 4 ln ∣ x - 1 ∣ + 7 ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x ^{2} + 3}{x ^{2} - 3 x + 2} : 1 - \frac{4}{x - 1} + \frac{7}{x - 2}

= \int 1 - \frac{4}{x - 1} + \frac{7}{x - 2} 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 1 d x - \int \frac{4}{x - 1} d x + \int \frac{7}{x - 2} d x

\int 1 d x = x

\int \frac{4}{x - 1} d x = 4 ln ∣ x - 1 ∣

\int \frac{7}{x - 2} d x = 7 ln ∣ x - 2 ∣

= x - 4 ln ∣ x - 1 ∣ + 7 ln ∣ x - 2 ∣

Add a constant to the solution

= x - 4 ln ∣ x - 1 ∣ + 7 ln ∣ x - 2 ∣ + C

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What is the integral of (x^2+3)/(x^2-3x+2) ?
The integral of (x^2+3)/(x^2-3x+2) is x-4ln|x-1|+7ln|x-2|+C