What is the integral of (5x+3)/(x^2+2x-3) ?

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

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integral of (5x+3)/(x^2+2x-3)

\int \frac{5 x + 3}{x ^{2} + 2 x - 3} d x

2 ln ∣ x - 1 ∣ + 3 ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{5 x + 3}{x ^{2} + 2 x - 3} d x

Take the partial fraction of

\frac{5 x + 3}{x ^{2} + 2 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

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