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

The integral of ((x+3))/(x^2-3x+2) is -4ln|x-1|+5ln|x-2|+C

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

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

- 4 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{x + 3}{x ^{2} - 3 x + 2} : - \frac{4}{x - 1} + \frac{5}{x - 2}

= \int - \frac{4}{x - 1} + \frac{5}{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 \frac{4}{x - 1} d x + \int \frac{5}{x - 2} d x

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

\int \frac{5}{x - 2} d x = 5 ln ∣ x - 2 ∣

= - 4 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣

Add a constant to the solution

= - 4 ln ∣ x - 1 ∣ + 5 ln ∣ x - 2 ∣ + C

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What is the integral of ((x+3))/(x^2-3x+2) ?
The integral of ((x+3))/(x^2-3x+2) is -4ln|x-1|+5ln|x-2|+C