What is the integral of (x+3)/(x^2-4x+3) ?

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

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integral of (x+3)/(x^2-4x+3)

\int \frac{x + 3}{x ^{2} - 4 x + 3} d x

- 2 ln ∣ x - 1 ∣ + 3 ln ∣ x - 3 ∣ + C

Solution steps

\int \frac{x + 3}{x ^{2} - 4 x + 3} d x

Take the partial fraction of

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

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

\int \frac{2}{x - 1} d x = 2 ln ∣ x - 1 ∣

\int \frac{3}{x - 3} d x = 3 ln ∣ x - 3 ∣

= - 2 ln ∣ x - 1 ∣ + 3 ln ∣ x - 3 ∣

Add a constant to the solution

= - 2 ln ∣ x - 1 ∣ + 3 ln ∣ x - 3 ∣ + C

f (x) = 3 x - 2 x, (1, 1)

z \to + 10 lim (\frac{1}{z - 10})

- 5 csc (x) + sec (x)

2 ln (3 x)

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

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