What is the integral of (x^2+1)/(x^2-3x+2) ?

The integral of (x^2+1)/(x^2-3x+2) is -2ln|x-1|+5ln|x-2|+x+C

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integral of (x^2+1)/(x^2-3x+2)

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

- 2 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣ + x + C

Solution steps

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

\frac{x ^{2} + 1}{x ^{2} - 3 x + 2} = \frac{3 x - 1}{x ^{2} - 3 x + 2} + 1

= \int \frac{3 x - 1}{x ^{2} - 3 x + 2} + 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{3 x - 1}{x ^{2} - 3 x + 2} d x + \int 1 d x

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

\int 1 d x = x

= - 2 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣ + x

Add a constant to the solution

= - 2 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣ + x + C

What is the integral of (x^2+1)/(x^2-3x+2) ?
The integral of (x^2+1)/(x^2-3x+2) is -2ln|x-1|+5ln|x-2|+x+C